If it were me, I would deny the animals and give them a stern warning about adding animals without previous written permission.

Denied by my daughter and roommates, of course.  

You cannot deny an applicant based on their assistive animal or charge pet fees/rent; However, Theresa is right.

With our already low inventory of SFR being available at affordable prices ( since nashville wages have not kept up with the cost of living and most cities have that issue so nashville is nothing special) if every new construction was pre-sold as an STR you have a few things happen,1) You turn area's or subdivisions in to hotels/ ghost parts of town where there are not occupant owners that want to live due to the issues assocaited with what luka mentioned2) Build quality is already low , but you de-incentivize quality construction because the it will just be a rental / short term home so the quality of the inventory produced goes down, so future investment and future inventory is hurt for SFR since not many people -- as being an agent, want to live in a home where it was used and abused for years on end, because a lot of times those homes doe not attract the average home buyer sine they want a home not a hotel and want better quality homes.3) you increase the rate of in-affordability if all homes were STR's .... in the past decade of living here what I bought when I first moved here at $127,000  in 2011 to What that same house is worth at $500,000 and I've done absolutely nothing to it except make it a rental property and maybe buy a few more with the heloc i got, but I am part of the problem of owners who bought, have multiple units , are not selling and contributing to higher prices and lower inventory but I am doing well for myself and my business which is fine, but i am in a position to say it's fine, where most people just want to live in a place they love and close to where they work, and if they cannot afford a home since all these STR are carrying an extra 150-250k premium on the sale price then it puts wealth building on hold or out of reach for others and government is the body that can regulate and but some guard rails on bothand finally then we have to consider what is best for the health of the city.

So I went to be pre-approved for my first investment property, but was denied by 2 lenders because the property would put my DTI too high.

We get these requests often and like you deny them. 

Ie do you think theres any chance they would deny the rental increase and opt out of paying? 

If the applicant gives you any excuse and then asks to work with them and that they can't afford what you are asking and they want a pro rata payment then deny them on the spot. 

A private lender denied me in the past due to my NAICS codes not matching.

I do have concerns that the repair may be due to abuse by tenant, but no way to prove it.