How to select a Lead In Fighter Trainer in the presence of both single engine and twin engine fighter jets? A major part of fighter pilot training is learning the handling of the aircraft itself. On the other hand, automatic controls on modern fighter jets take away a lot of the pain and hassle. I am curious about the extent to which such ease of use plays a role in choosing the Lead In Fighter Trainer (LIFT) platform for an airforce.

Usually, it is assumed that the manufacturer supplying fighter jets will also supply the LIFT platform. But an airforce could buy jets from multiple vendors. It would be extremely cost ineffective to buy a different LIFT platform for each type of fighter jet. So airforces should try to standardize the LIFT platform they use.

Assume an airforce buys both single and twin engine fighter jets, from two different manufacturers. Neither aircraft has a twin seater version, so the first time pilots fly in them, they don't have an instructor sitting behind them giving instructions. Obviously, the aerodynamics and general handling of a twin engine fighter jet ought to be quite different from a single engine fighter jet. Now in making a decision on standardizing the LIFT platform, would this airforce take into account the presence of both single and twin engine fighter jets in its inventory? Would it procure two different LIFT platforms, one for training pilots towards single engine fighters, the other for training pilots towards twin engine fighters? Or does it make any practical sense for the airforce to choose only a single engine LIFT platform, or only a twin engine LIFT platform? What would be the pros and cons of such a decision?
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.