TK cannot force UA to open award space, but it can make the request through its *A liaison system. That typically takes at least a business day. If you are getting an immediate "no" that means nobody is making the request. Call back and make the specific request that TK's *A liaison reach out to UA. If you get the verbal equivalent of a blank stare, that becomes one more route to speak with a supervisor (recognizing that you have tried in the past).

If that does not work, the next step is a three-way call with TK and UA. UA has no obligation to help and cannot reissue TK's award ticket, but UA can be decent in these situations and may open the space on your request at which point the TK agent should be able to see the space and reissue the ticket.

All of this is work which TK should do, but if it won't, you are stuck doing it.