Birdman isnt “broke” by regular people standards but he damn sure isn’t living it up. And this isn’t some sort of complex plan; he doesn’t have the money to pay for it. Somebody whose whole image and persona is about being rich isn’t going to have a story about his house (his main house too) being taken away from him go public.
Same with paying Wayne and everyone else. If he had it, he would pay it. That money train dried up a long time ago so he’s trying to keep what he has.
The cheapest, richest people in the world still pay their mortgage/rent. There’s no long term financial strategy in defaulting on a house. There’s a difference between not paying for a service you already had done, and not paying for a house.