A FLAT LIE that's your property, not theirs, At this point make them put it in the trunk unassembled (As it stands RIGHT NOW) and before they can "Accidentaly" Overhaul it, They've fucked you out of $550 right now, and if they do ANY more work the price will Jump to 3 grand or more.
Unfortunate that you let them disassemble it, Now they GOTCHA, the "Remanufactured" part they want to install IS YOUR TRANS
If you can, show up at their shop with a cop or a deputy and INSIST they put it in the trunk, they'll squeal and balk but DO IT them find a decent trans man (Perhaps the other AAMCO?) to see what's really wrong with it (If anything)
As it stands right now You're going to need somewhere around 3grand to "Fix" it, DON'T LET THESE THIEVES PROSPER.
Do NOT give them "Permission" to procede (And be careful, they'll trick you into saying something like "Well If it HAS to be overhauled" THAT'S PERMISSION TO SCREW YOU)
You're in trouble, right now they've got your car as hostage, GET IT BACK.