What you can do in 1h 55min in IST is, if the TK agent in Varna refuses to interline for you, to run to the transfer counter with your onward BPs. The transfer counter staff are more sympathetic from my experience. They might help you call the gate/baggage handlers on the ramp to find your bags and send them to plane for your onward flight.

Just make sure you don't put anything of significant value into your checked baggage.