Its a tricky situation. Obviously you are not being unreasonable to be concerned about her but you really need to think carefully before trying to 'fix' her life for her. Firstly because it will come across badly, she'll just feel that you are criticising her which will make her feel even worse and possibly exacerbate comfort eating. Secondly it won't work, she has to make the decision for herself that not only is she overweight but NOW is the time to do something about it.
I think you need to focus on the health and happiness side of things rather than mentioning her weight at all. Its not about how she looks, or what the scale says, its about her feeling good about herself and doing what she can to minimise the symptoms of PCOS. Has she had her thyroid function checked? I have had similar issues to your sister and I am hypothyroid. This is easily treated if it is the case for her.
I've always felt big and my eating has been hand in hand with low self-esteem. Though, unlike your sister, my family also struggle with their weight.
I put on a lot of weight in the few years after I left home. This was partly because I was able to choose my own food and pick things that had been rare treats at home. It was also partly because I picked up my DP (now DH)'s eating habits. He's not anything sinister like a 'feeder' or anything. Its just that he also has unhealthy eating habits and we both felt down a lot and used food to (temporarily) cheer us up.
I grew up in a big family. This means I can be a bit competitive about food (wanting to make sure I get my 'fair' share, not leaving things till later incase someone else has them, etc). This caused a big problem when I moved in with DH because although he isn't gaining weight on the amount he eats, if I eat the same as him (seeing it as 'fair') I gain weight. Its partly because women have lower calorie requirements than men, partly because I'm less active, and my thyroid probably doesn't help.
I eventually got to a BMI of 53. I had fertility problems too and we wanted to start a family. I knew for years that I was overweight and that I wanted to lose weight 'one day'. It doesn't matter how many people tell you that you are overweight, indangering your health etc. I knew I was. For me, depression was a big part of it, I was so desperate for any short-term comfort that I ate even though I knew it was making me unhappier in the long term.
I don't know what changed exactly, but I realised that I couldn't keep putting off 'one day' if I wanted a family (tick tock biological clock). I started tracking my weight on the wii fit and by cutting out snacks and reducing down my meal portion sizes I got my BMI to 39. Then I got pregnant. I've tried to keep my weight gain down in pregnancy and I'm looking forward to getting back to losing weight after baby is settled in so that I can get down to my target and stay there. Because when I was losing weight, I realised that not only did I want to have children, I also want to be an active parent whose a good role model to them.
I guess I'm trying to say two things. Firstly, its unlikely that her partner is evil. He's just got the same problem she has around food, but its affecting him less because as a man working a physical job he can get away with eating more. Yes he should be more supportive when she is trying to lose weight, but he's more likely to support her if the request comes from her. Secondly she needs to find her own motivation to do it, because its bloody hard, its changing your lifestyle forever (short term diet mentalities don't work with someone that overweight because even if they hit target, they then 'finish' the diet and go back to the one that made them overweight in the first place.
I really recommend that you avoid mentioning her weight, her diet etc unless she specifically asks for advice. She knows what she needs to do already and (however nicely you do it) it will just come across as criticism and dent her self esteem even more - resulting in more comfort eating.
Instead? Value her for who she is, and help her see it too. The good news is that as you don't share her eating habits, your company will be a positive influence, because you won't make everything you do together centre around food. Plan some days/evenings that are about something other than food, its okay if they include a light lunch. But even better if they naturally limit eating opportunities and involve some activity (doesn't have to be something obvious like a sport, can just be walking around an exhibition etc). If you are going to try something obviously sporty (dancing, swimming etc) then make sure that you don't frame it as 'I think it'd be good for you to do this to lose weight' instead try 'Ive always wanted to try x, I'd love you to come along too for company'.
If you show her that she's someone worth spending time with, then hopefully she will value herself too and start caring about herself again. Even if it doesn't work out immediately, at least you'll have more time together :) Keep trying, if she's shy and self concious then it might take her a while to respond.
Its great that you care but its a minefield so take it slowly and tread carefully.