Hello FaithfulTiger, it's good to talk to you.
Firstly I want to say that I'm proud of you for talking about something that is clearly difficult and I would imagine is hurting you inside, because I found a beauty in your words not a tragic or pitying quality.
I also want to say I empathise with you about your Father's passing; I can only begin to imagine how that impacted on you and your sister, knowing nothing about him or what he was like, not that I'm asking, as I feel it's not right to pry, only your right to discuss such things if you wanted to
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I also want to tell you that you are not being selfish. You care for her and the fact she is around 550lbs in weight, is actually to be honest in a respectful way a selfish way for your Mum to live, because it's not considering how others around her might feel if she gets ill, so you're not being selfish at all, because you care and I could quite easily believe that there's times where you find her behaviour upsetting, hurtful and confusing and that would be understandable, because you're human and you are an emotionally beautiful person with a heart and feelings, that much is abundantly clear.
The undertaking you have been going through has clearly been tough and there is a problem here with the family dynamic I see, as your sister may be countermanding your requests to not panda to your Mothers obsessive food relationship, by what I can gather and as you can't be around the situation all the time, then it's clearly difficult for you to make any meaningful long-term impact on her circumstances, unless your sister can keep up the things you try and encourage your Mother to do in your absence.
What I would say at this point is, definitely do not think about surgeons intervention of any kind. I deplore such practices, as it ignores some of the possible non-invasive options for the individual, by focusing on the previous options not the unused ones that could be used and then surgery puts them at risk from complications and possibly even a fatal reaction to the anesthetic and whilst there is a still a chance for her to lose weight via many differing options, no interventions should even enter into yours or your sisters thoughts
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The main issue as I see it is your Mother is showing an extreme reluctance to want to change and to be honest, badgering her may not be the right emotional tone / stance to be adopting, but you may want to try a more persuasive stance with her, perhaps try to rationally help her to see that losing one parent was no doubt hard for you and you don't want to lose another so soon. Let her see how much she means and how much you might mean to her and see if there is a way to make some gentle controlled expression of emotion find it's way through to her, because being highly negative, like arguing or shouting, I'm not for one minute suggesting you do that, wouldn't be the way, I feel, to connect with your Mother on this issue, but persuasion through recognising that a potentially serious and emotionally painful future for you and your sister may occur if your mother won't act on this, might give her the gentle jolt she needs to accept yours and your sisters help.
If you have not done so already, perhaps discuss with her how weight gain makes her feel, for example when she has had to go up a dress size, does she see it as nothing to worry about or embarrassing? Does she relate any negative thoughts to aspects of weight gain and if so, how does the weight gain make her feel negative and see if these realisations might spark her conscience.
You could encourage her to do gentle exercise, but it would require your sisters help.
For example, you do small amounts at first like walking approximately 20 feet from a chair and back again once a day, whilst you or your sister hold her arm if needs be as a comfort blanket for her in case she fears falling or tripping over.
Give her praise every time she does the walking to make her feel like it's an accomplishment of some kind, not some sort of punishment for being extremely overweight.
Then the following week, double the distance per day and every week, add the same distance on, so after 4 weeks, she's walking 80 feet and then after 8 weeks 160 feet a day.
After 4 months, she'll be walking 640 feet which is just over 1/10 mile a day..
Within 8 months hopefully this would equate to 1/4 mile a day, so it's then small regularly controlled bursts of exercise, nothing too demanding, just gradual increases per week.
As for the diet, try to at first plan a system of meals that works, but structure it around the current calories, so as not to cause crash dieting, as the one thing you don't want is for her to be losing significant amounts of weight every week.
Usually due to a loss of a fairly significant amount of salt in the diet, weight loss for many can be quite rapid in the first 2-3 weeks, as water is lost as well as fat, but for many the weekly amounts then tail off.
However If people are losing 8lbs in the first week for example and it's still happening 6-8 weeks later, then that's an issue as the persons body could be scavenging muscle and that can cause metabolic problems.
What I meant before about structuring meals around the current calories was to base a plan of attack not suddenly chopping a load of calories out of the diet as your Mothers body would potentially create severe starvation type feelings, and feelings that would be extremely hard to suppress, therefore subtle changes in caloric intake would be recommendable, so the body doesn't get everything it's used to, but because it
initially it would be getting a bit less, it would have less reason to grumble.
So to begin with, let her have 6 meals a day evenly spaced out, so the eating stops being random and "whenever", if that is the case. This way her eating behaviour can be better monitored and defined.
Secondly, try to make subtle changes, I:E: if you use cooking oils to cook any of her meals, use monunsaturated / polyunsaturated ones, not the normal ones and then regulate how you cook with them.
So if you compare 1oz of lard versus 1oz of sunflower oil, the difference is around 80 calories, but the fat is better. If you did this 3 times a day, you'd drop 240 calories a day, but your Mothers body wouldn't be complaining a lot as you'd not be ditching let's say 1,500 a day.
This would work if for example you cooked beef patties to make homemade burgers, or you fried eggs using a little fat like lard or a solid fat substance found in packets or tubs versus liquid fats in bottles.
Other simple changes might be skinless chicken versus normal. Or wraps from a shop that might have high fat dressing on them like ranch, so you ask for them without the dressing, then when you're home add 1tbsp of light ranch onto the wrap, which means you're losing around 30 calories from the wrap.
If she eats yoghurt, go for low fat instead.
What you might want to do is work out roughly using something like Nutrition Data or About.com calorie counter to work out roughly how much she eats per day, then make subtle changes so you're only getting rid of say 250-300 calories a day and after 2 weeks cut back again with some more subtle changes and / or slightly reducing portion sizes.
If she eats pizza from a store, work out the calories, then buy a pizza kit, that has let's say 100 less, because then you're getting rid of 200 calories a day from the diet if she eats 2 pizzas a day.
This softly softly approach to food should work better, rather than just ditching a pizza and forcing her to eat some kind of chicken salad with nuts on it, where she might object strongly, but if you're keeping some of it the same, but altering how it comes I:E: slight smaller portion sizes, or using light dressing not usual or light yoghurts not usual, or making stuff with healthier fats, then eventually if you could even get her body to adjust to consuming 500 less calories a day after 4 - 6 weeks, without it causing her noticeable starvation type feelings, then losing 3,500 calories a week, can reduce weight by 1lb a week, but if she loses a bit to start with possible through water weight from reduced salt intake and also from the gradual exercise, that will help.
It won't be easy and if you get 10lbs down by 4 weeks and then have to settle for 1lb down a week for some time, then that might be what you could aim for, but you'd be getting somewhere and not causing a massive problem for her with starvation type feelings and quick weight loss, caused by drastic calorie cuts that would make her unhappy and more prone to potentially being cranky and demanding more food.
I really hope some of these suggestions might work, but it will require vigilance and a little changing of how you and your sister are around your Mother RE: buying food, or doing certain things that may panda to the weight gain and whatever happens, don't give up hope and please do keep us updated, or if you need to ask anything else don't hesitate to ask.
You have done the right thing by bringing this up and it's clear you care and I am proud of you for talking about this and you're not alone with this okay
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It's almost like a alcoholic the only person that can help them is themselves. but at the same time not sure if you should be an enabler either. 
