Really good points by Ltl.
I've been noticing this in my brother recently, although he's great at Squatting (200) and Pulling (230) his Bench is really lacking. Part of that is to do with bodyweight, he's only around 75kg but also he's finding it difficult, as you are, to transition from 'manageable weights' to '**** that's heavy weights'.
The Bench in particular is so massively determined by your body position, if you can get your body position correct, keep your arch, tightness, legs tense, upper back pulled back and all the rest of it you will find thing a lot easier. If you don't then like a loose string the whole deal unravels and you're left with just pure pressing power to press that barbell back up and at maximum weights that just won't happen.
Mobility is great, also just concentrate on shrugging back strongly for as long as you're on that Bench. Your body should actually raise up, from the rearward shrug of your upper back muscles, you want that much of a contraction.