The way not to be a pushover is when you have a stronger reason to stay on the course than to give in.
In this case, if you aren't sure whether the BIL is the right choice, then it would be better to keep the bird and see how committed your BIL is to Jake. Explain to him that you feel a responsibility to Jake and after all Jake has been through you want to make extremely sure that whoever he ends up with is the best person for Jake. Say that by coming to your place and interacting with Jake and also learning from you, it will give you greater confidence in deciding whether he (BIL) is the right person. Also whether Jake just like him for a short while when the novelty is new (hey, if people can get birds and enjoy them while is is a novelty, why can't the birds do the same to humans)?
Hey, wait a minute, birds already do that. After all, don't some of us have experience where the birds love us initially until later they find another person to love and the 1st person is dumped.
Of course I've not put it so 'tactfully' but you need not put it too tactfully. Why? 'Coz 1 of the qualities for a good bird person is when he is able to take opinions which he may not like so well for the benefit of the bird. That means that he will listen to advise that is not just about 'making him feel good'. That he loves the bird more than 'the need for someone to stroke his ego'. That he is able to see why people are asking those questions or saying those things and not misinterpret them to their liking
Just my 2 cents
Edit to include:
Your heart says Jake is better off here. Maybe its your instinct/experience that is telling you that for the time being (maybe temporarily or permanently) that Jake is better here. Why not explain to him that its not a permanent no to BIL but currently you are evaluating whether he is the right person. After all you are now in the fostering and rescue biz.