Just tell them it appears it's just not a good fit for their needs, and now that they've told you they intend to let children play in an unfinished basement, that you can be liable for anything that happens, so you are denying them.Whenever it's this much work, just deny.  

Why not deny for the eviction?

Side note:  Obtaining a judgment from small claims court would only be beneficial if the tenant has some sort of asset (i.e. owned a car/home/employment) that you could go after if they continue to deny responsibility. 

I have been building my credit up and I'm around 700 now but when I tried for a refi i was denied.

All things I normally denied for - relative for a reference and bad credit.But, she offered her mother as a co-signer, who no doubt wanted her grown daughter and the kid out of her house.So, daughter moves in, mother pays the rent on time.  

The only benefit for me in her being a support animal, is that I don't have to pay a pet deposit.If it was me, anyway, I would deny based on too many animals, that you only allow one support animal.

It also makes it easier to deny them when they don't meet these qualifications. 

And give them a fax number to fax it back to me.Then, if the applicant actually provided this, I'd Google and make sure this medical fax number is from a bonafide medical office.And don't be intimidated by a tenant who threatens to complain to HUD if you do this and deny them.  

It might not be worth the legal battle to deny them.  

Deny.