Back to OP's original question - there are 2 handling agents in IST - Havas and Celebi (plus TGS that handles mostly TK). All have transfer desks opposite transit security/passport control. These are huge - you cannot miss them. Shall be open 24/7.
As the OP has pointed out he's not in transit, he's on 2 separate tickets. Arriving IST on TK on day 1 and departing IST on BA on day 2.