The mounting piece (which is part of the door liner - about a 1/2" stud/tab) broke off and the butter compartment cover fell and broke. While I can find a replacement for the butter compartment cover, what can I use to reattach it to the door liner?
Not being there all I can do is give a suggestion and you will have to determine if it will work in your situation.
Get a screw of the correct size and length.
Drill a pilot hole in the center of where the old stud was.
Then thread the screw into your pilot.
Then remove the screw and cut the head off, A Dremel does a nice job.
If the end of the stud is where your hands will touch it clean it up so you do not get scratched all the time.
Now just use pliers to re-install your new stud to the correct depth.
Depending on the set up you may not have to bother with making the stud just use the screw and install it through the hole in the new cover.