@Gary Dezoysa
I'm an landlord/investor and my wife is a property manager. I hope our insights help...
I don't think most professional property managers would be open to this stipulation for a couple of reasons.
1) PMs want to make sure that everyone is getting paid. In my experience the PM collects rent (usually at the beginning of the month), pays vendor invoices, refills the maintenance reserve account (if there is one), pays themselves the agreed upon management fee, and then distributes the owner's proceeds. This process guarantees that they are getting paid what they are due, but also that the vendors and contractors that they worked hard to establish relationships with will also get paid.
2) PMs who want to grow their business need to keep processes and procedures consistent across all of their different property owners and properties. Being a PM is a very time consuming job. Deviating from their procedures will likely take time away from more important/urgent matters.
With that said, a PM may be able to help with rent collection in other ways. For example, a PM may offer paynearme as a way for tenants to pay rent with cash at their local 7-Eleven. Interview some local PMs, see what they say.
Hope this helps!