Screen and reject the same.

The bank will just accept, reject, or counter. 

They procure three bids, they reject two of them.

Some will get rejected by 2 or 3 people and give up.

If you can't put your reason for rejecting an applicant in terms that you apply to every person you are vulnerable to a suit due to violation of equal housing laws.

You likely will have FAR MORE rejections when trying to buy from the MLS dealing with an agent, HOWEVER, I see all the time properties that are really over priced, no one submits offers and finally the seller excepts your offer because they've not had anything OR you caught the at the right time and the seller is highly motivated.  

If you do not allow them, you lose out to half the renters in the market or the renters will just get a fake prescription for an emotional support animal and you'll be forced to allow the animal but not allowed to monetize it.Instead of rejecting animals, figure out how to protect yourself while monetizing it.Personally, about 1/3 of my rentals allow animals and we rarely have a problem that's not covered by the security deposit.

Shop banks, CUs, Brokers, Private Money, etc...First, have the courage and dont fear rejection.

Because lenders will look into your financial situation - the combination of low capital and bad debt will most likely have your loan application rejected.

Realistically youre offer doesnt matter as to the price, as the homeowner will accept or reject the price...but that has no correlation as to whether the bank will.