What is the best solution to this problem?

I don't like their interface very much - I do not find it intuitive - but they're the ONLY platform I've found that truly understands the nuances of STRs and direct booking and provides solutions for basically any scenario (I could find other, slicker, more expensive platforms that would do everything BUT... automatically apply a damage waiver to every reservation, which is something I very much want for direct booking).  

Any advice on tapping into the creative mind and finding a solution to this issue? 

I own and self manage 6 duplexes).Solution:- Keep building the relationship (pick your battles :)- Property Management ($$$)- Lock box for each property (to leave a spare key in), for contractors that you already trust. 

@John Teachout We both agreed that a month-to-month addendum would be a good solution.

First step is realize your need for a home for your family doesn't negate what land lords/PMs require to rent to you.Second step and a possible solution is to the problem is put yourself in the LL/PMs shoes so you can understand why they have that requirement. 

To say this differently, the minute you suspect they've given up or are stringing you along, then yes....lower the boom and issue Pay or Quit.Here's one solution to getting paid: put them on weekly rent payments until they recover the full amount of their payment. 

This is fine and it is a viable solution but you have to choose the management company via extreme vetting.

@Thomas Meyer  I would like to do a more cost effective solution like that next time I have a vacancy, but these people were so determined I'm not sure if that would be adequate.

I fully agree with @Nancy Bachety and I like the concierge solution suggested by@Anand S.