Getting back together with the girlfriend would help his emotional state and his financial state. It would also seemingly be what she wants too so... What's the downside here?
Don't go saying it like all relationships are destined to be some rocky boat in a stormy sea, some people are actually happy you know.
And well, solving those two problems are really step one MC should fix before BEFORE taking in his "sister" if possible, and it looks hella possible.
How would she help his financial state? By lending him her parents' money?
The downside is possibly on her. Unless he can recover, she would go down with him. While their relationship isn't destined to be some rocky boats, his boat WAS and still IS rocky. Not everyone has the heart to invite someone on a rocky boat, let alone a potential sinking boat. He was in a situation where he can't be optimistic. Unless he's fine feeling like a parasite, starting a relationship in his situation is hard.
And while she said she wanted to make him happy, she hasn't done anything concrete as far as we can see. If she was truly fine with a rocky boat, she should have jumped in regardless of what he said, like ctually do something to help him. Maybe then he would have more confidence sailing with her.
I know some people are happy. But is he?
Yeah, ideally he needs to stabilize his life more before helping his sister or starting a relationship. But he probably thinks it's better to save her sooner than having her experience the same thing as him: losing family.