PA-28 Flight Instructor reported a runway incursion occurred as an aircraft that was landing did not comply with ATC's instruction to exit the runway at a specific taxiway. To ensure there was proper clearance between aircraft; the Flight Instructor requested a different taxiway to safely exit off the runway.

Narrative

Student training flight. Landed full stop on Runway XX and was instructed to taxi via Taxiway 1 and hold short Runway XY. After holding short for several minutes we were instructed on Tower frequency to cross Runway XY at Taxiway 1 and contact Ground on the other side. Before moving we looked left and saw a Lancair who looked to be landing Runway XY so we held position. As soon as Tower gave us that crossing clearance; he said to cancel the clearance and hold short of [Runway] XY at Taxiway 1; to which we read back we saw the landing traffic and would continue to hold short. The next clearance we received was that we were cleared to cross [Runway] XY at [Taxiway] 1 and the Lancair would be exiting at [Taxiway] 2 off Runway XY. As we pulled forward we could tell the Lancair was not going to take the exit he was instructed to so we stopped the airplane on the west half of the runway and told Tower that we were going to get off the runway via [Taxiway] 2; reverse high speed exit. The idea behind this was that the Lancair would clear the runway at [Taxiway] 1 but because we would be behind it we would be unable to clear the hold-short if he was stopped off of [Taxiway] 1 and Taxiway 2 intersection. The Tower quickly approved our taxi to [Taxiway] 2.The airspace was very busy at the time with lots of training aircraft and more helicopter training traffic than I ever seen in my time at ZZZ. I believe the Tower was task saturated and in the process of having a control on another frequency to control the west side of the airport. The Lancair pilot read back the taxi instruction but did not comply with the instructed exit nor did he immediately get on the radio to say unable or anything of that sort; which played a contributing factor. Luckily we saw him and could tell he wasn't going to make the turn at [Taxiway] 3 but had it been a jet I dont know that we wouldn't have had a collision with larger wingspan and much faster response time requirement; closure rate; etc.The ZZZ Tower folks are a great group and usually have a very good handle on the air traffic.