I've earned points for a direct billed, long-term, Residence Inn stay (4 months at one property, 5 at another); so I know its possible. I didn't have to do anything special other than give them my MR number when I checked in. This was also on a specially negotiated rate (albeit only $5/night better than available on marriott.com).

If nothing at the local property works, I'd suggest asking for a copy of your room folio and then sending it to Marriott directly for credit. If the manager or daytime desk clerks aren't willing to give you a copy since it is direct billed, go a 5:00am some morning when the night auditor is still working the desk. They'll give you a copy.