I have not increased their rent (although I should have).

You would ideally want to buy in the downmarket for below market price (around this time of the year since its more of a buyers market) and then sell in spring when values are going to be increased and also have a sellers market (you could sell for above market price).

If it were zer0 dollars out of pocket without increasing the loan amount, it seems like a no brainer, but increasing the loan has me skeptical.

There are going to be three levels of addressing this 1) things you can simply go there and do anytime (pick up pet waste and throw it away) 2) things you can do given proper notice (enter the unit and inspect, repair damage etc) and 3) things you can only do given proper process/procedure (notice to vacate, increase rent etc.)

I double "co-hosters" in most markets are getting 25-30%.If property management interests you I would suggest you become a licensed and insured PM company to increase your credibility. 

Is KNOWINGLY increasing a payoff to an amount ABOVE what was technically and verifiably owed illegal for the lender and or borrower to do?

This tends to occur when the appraiser is unfamiliar with the area, and the area has had a recent rapid increase in property value.  

Permits are now required for all STRs, with fees increasing as of June 2024.

It's always a plus when you can get your property valuation increased and the tenant pays for the new item, just so long as your downside risk is mitigated, too.Good luck!

You can charge an increased deposit, pet rent, or require the tenant to carry pet insurance.