CELTA TBI tends to be somewhat less expensive than those provided elsewhere. The difference would go quite a way toward covering your living costs. Aim for accommodation in the Kuningan area. You don’t want to be dealing with too much traffic. Generally, it’s just a nice area. As far as possible, don’t skimp on your lodgings (during the course, at least). 

If you’re not totally set on the idea of working for a TBI in Jakarta there’s every chance that you’ll get an offer on employment at one of their other branches. And it doesn’t have to be TBI, does it? 

I won’t rattle on about any of this. In short, I think it sounds like a decent idea well-posted. Good thinking! And good luck.

Fatmawati is in the far south of Jakarta and as far as I’m aware there are plenty of accommodation options available. If you decide to do the course there, I’m sure TBI will be able to recommend some places to stay.