1. You do now want to deal with TK IST Customer Service agents - believe me (and I am their highest tier FF - and I am mean it)
2. Once you are checked in, UA will not be able to change the ticket as TK segments would be in the Checked-in status. UA will not be able to offload you using their SHARES system, this needs to be done from TK Troy system. TK would have to offload you first (if you have luggage they will have to locate it and deliver to you), and only then UA (when the segments are in Open for Use status after offload) would be able to reissue the ticket. Now look at point 1 and ask yourself how likely it is for this to happen.
My piece of advice - just fly to BUD as planned...