MR16 LED Bulbs - Fitting too narrow and replacing old Halogen bulbs
It's common for LED MR 16's to not fit the same as their predecessors.  Those types have the AC to DC rectifiers built in and work with most any 12 volt supply.  Those cans probably have a small 50VA magnetic transformer attached to it inside the ceiling for each can, verses one big gigantic transformer for all the lights.  

As for the gimbal not fitting back in the can, sounds like you need to find a different LED bulb instead of trying piecemeal together multiple parts.

Your trims don't seem to be from a popular vendor such as Halo or Lightolier. Your best bet is to look for some part number and google it.

If you're still hard-pressed on replacing the trim kits and not the entire fixture, you might can buy a 4" GU10 trim kit from the local hardware store and use the MR16 bulb instead of the GU10 bulb, and maybe, just maybe it will cover the existing hole and/or not fall out from being incompatible.