I don't want to get into the specifics about how this happened. Will the compression be too high for this to work out? Has anyone else ever done this before or am I out of luck?

Edit:
Forgot to specify that it's an 84 22R.

 

Do you have details on the differences between the two pistons?
You need to know if the top profile is different, what the deck height is, what the bore is, has block been decked, compressed head gasket thickness, and you can calculate the comp ratio. Assumes you have no physical interference between the piston and head.

was it a stock rebuild and you just bought the wrong pistons?

i know guys used to run 20R heads on 22R shortblocks for higher compression, but dont know about pistons.

 

If your head has not been installed yet you could put a tiny ball of Play-Doh or clay on top of each piston, set your head down on the block with all of the valves closed (no cam in it) or properly timed with the chain on and a couple of head bolts finger tight and roll the engine over to check for interference. I have no idea about what the compression would look like.
If the "bad mechanic" wasn't you make him fix his mistake or try putting it together and run a compression test.

 

The flat top piston is the 85 and up, using these in place of the older style piston is going to significantly drop your compression ratio.

Trying to put the dome piston with a flat bottom head or on a shorter engine block is going to cause a collision, bend connecting rods ect.

 

I bought my 4Runner and drove it 400+ miles home with the wrong pistons in it. It was under powered and ran rich but it also had other issues that contributed to that fact. I replaced the motor with a hybrid 22r soon after. I wouldn't keep the wrong pistons in there though and would make the mechanic replace them and make them pay for it.

 

If you or a friend are the mechanic then pull it apart and put the right stuff in. If it was a pro shop make them make it right.

If it was you no biggie. We all make mistakes and correcting them usually isn't to bad.