Since you haven’t gotten any answers from an actual plumber yet, I’ll give it a shot.

There is no linkage to a trip lever, as your tub is setup with something similar to this style of stopper: https://www.amazon.com/Kingston-Brass-DTT201-Tip-Toe-Polished/dp/B003E2NUZS/ref=mp_s_a_1_17?keywords=pop+up+tub+drain+stopper&qid=1659560528&sr=8-17

I think this video will be helpful to show you how to make the repair: https://m.youtube.com/watch?v=xrOIGlPCgtw