How to select a Lead In Fighter Trainer in the presence of both single engine and twin engine fighter jets?
Fighter lead-in isn't about becoming an expert in the training jet, it's about learning the fundamentals of flying tactical aircraft -- air combat maneuvering, strafing, bombing, etc.  So the difference between the aircraft flown in LIFT and the aircraft that the pilot will go to next isn't all that important.

I've never heard of any issues the US Air Force had with students flying AT-38's (2 engines) in fighter lead-in training then going to F-16 training.  The unique aspects of operating the single-engine F-16 are all covered during the F-16 course, and there are LOTS of differences between that and the AT-38.  Removing one of those differences, the # of engines, wouldn't be worth adding a whole new aircraft to the LIFT course -- a very expensive proposition for negligible gains.

So no, the number of engines in the eventual aircraft vs in the LIFT trainer is immaterial.