I'm so sorry to hear about your sister but it's really positive that you are there for her - a lot of abuse sufferers feel cut off from those who really care about them, so it's really nice and reassuring to see she has someone like you in her life.
This is quite a difficult situation and must be frustrating to watch on the outside. You mentioned that your sister is too tired and exhausted to fight the issue and it is easier for her to just accept what is going on - I'd say theres more than financial abuse and and his emotional abuse has worn her down where she is resigned to her situation.
At the end of the day, the decision to leave has to be down to her and the impetus to change is down to her but there is alot you can do to help build this up and you've already made the first move by acknowledging what is going on.
Think about how to empower your sister and build up her confidence and self esteem. Look at building up and strengthening her social network if you can - this may not be possible with him present but you are a huge resource for her. Look into how she can siphon off some money to help leave without him knowing, can you and your family save together in a wee fund for when she leaves (and make her aware of it?) so she can get back on her feet. Do you have a place where she and her child can stay until they get settled? If so, let her know that. Try and go out with her and do things she enjoys so that she knows there is enjoyment outside of her life with him. This bit is also difficult as he is controlling her funds but maybe you or family could help out a little (ie dance class, socialising over lunch or coffee, etc).
Your sister has very limited resources atm and he seems to be taking everything she has, we want to build her up so that she realises she doesnt need him as a resource and has others around her and then build on to working on herself.
One of the main things people working with victims of domestic violence look at is working with them on identifying what makes a good and bad relationship. I dont think your sister is anywhere near ready for this as she is still with him and stuck in her situation- but this is definitely worth looking at when her confidence and spirit is higher.
Apologies there is no quick fix but I think the above would definitely help in empowering your sister and giving her the confidence to leave. Fingers crossed for you all!