Not familiar with the area and rentometer has no data i can find on rent rates.

Investment property equity lines and second mortgages are out there but generally also require full income verification (including rents.) - Speaking of HELOC's and second mortgage another option could be a simultaneously closing.

Whatever rent income/expense we receive/incur, is it split the same % as well?

I connected the seller with agents in other markets and also set up an MLS search in case they wanted to rent instead of buy upon selling.

What's a reasonable amount of time for a property management to tour, provide quotes, and get a property advertised for rent after a tenant moves out?

That has led to inflation, which has led to the Fed's hand being forced to raise interest rates. 

What rent are the tenants paying, and is it at-market?

(Like 2 years of verifiable rent payments) as an example.  

If you prefer a private space, you might also look for renting a meeting room at a hotel.

TCU area is very tough rent/price.