The IMA light isn't coming off like it did an autostop, right?

When you switch gears you are probably letting off the gas. Is it possible that the engine is somehow stalling out, and then when you go back into gear, reengage the engine and press on the gas, you're actually bump starting the car? In other words the problem could be as simple (or as complex?) as something causing the engine to stall for some reason.

Obviously the engine shouldn't be stalling, but I wonder if that could explain what you are experiencing.