You could use the security deposit to cover May rent if there is no damage; otherwise, deduct the damage, then apply the remainder to May rent and inform your tenant they owe the remainder.

I also don’t charge extra pet deposits or pet rents and this makes my rentals super desirable to people.  

You're being encouraged to finance people's earnest money deposits.

What you do if you move forward is not a new lease but estopel agreement they sign, your lawyer would prepare that should include the terms of the agreement that it is month to month, the amount, any deposits, any appliances etc owned /not owned by tenant and and special terms ( for instance how do you know only 2 vehicles are allowed).

take double deposit or more if you can and a top of market rental rate.

I can add a statement clarifying that the payments are guaranteed regardless of occupancy, but I’m unsure how to handle the receipt for the deposit and first month’s rent in this nontraditional arrangement.

I am helping investors who paid deposits on Bennah Oaks through Wagner/B2R.

It collects our rent, applies late fees, let’s tenants report maintenance issues, deposits our rent into our bank account the next day, screens  our tenants, markets our properties to something like 25 websites, has full accounting and reporting and a whole lot more.. 

So, all you can do is offer them a "carrot" or a "stick".Carrot - Cash-for-Keys or security deposit returnStick - threat of small claims case and garnishment (if they have a job, bank account, etc.

I give them a deposit for the job once I close and from my understanding permits were to be submitted right away.