You should expect lines at the airport to board Magical Express (ME). The coaches continuously load, first come first served. Some coaches are shared between resorts. Worst case, expect multiple stops to other hotels before arriving at your hotel (MK resorts sometimes share one coach to 4 hotels). Then it takes a few minutes to checkin. We try to allow 1 1/2 hours minimum to get to the hotel entrance, not counting the checkin process. Then you will have a wait for transportation to a park, which may come as seldom as every 20 minutes or longer.
So, no, I don't think you should reasonably expect to be at a restaurant in HS within 2 1/2 hours. You *might* hit things just right and sail through the process, catch a bus as it's leaving and be the first stop. You could then go straight to park transportation and skip checkin (assuming you have no carryons). In that case, you *may* have enough time.
You most certainly could make it with a rental car and no stop at the hotel to drop bags.