Tail,
61.157 of the FAA regulations says that for a type rating, you must have a record of ground and flight training. So no training center can just let you fly the sim for the checkride. You can do the the SE or ME ATP without training if you think you'll pass but most people want a few hours warm up for the checkride even in a light twin. Your ATP written tet is good for 24 months. If you pass a ME checkride in those 24 months, you'll have a FAA ME ATP and you can add whatever type ratings you pass the checkride for whenever. If you take the checkride in a SE airplane, if you want to add a ME rating or type rating in a ME airplane after July of this year you'll have to go back and complete an ATP training program and pass a new ME ATP written.
My advise would be to complete the ATP ME checkride within the 24 month window. If you do it as part of recurrent training in the jet and get the type, that's great. If not, do it in a light twin. I have a feeling it's going to become a nightmare starting this August.