Made a Mistake on on of my kits and glued the wrong part in the wrong place

I have heard of ways to break apart the pieces without damaging the plastic, but I am not sure what I should do

I am using Tamiya Extra Thin Cement and need to fix this or a $50 kit goes down in flames

Unfortunately the glue in question, actually melts the plastic. Your best option is to cut piece off with a good razor saw or hobby knife. Then just re-glue in correct position.

That's true but you have another possibility. You can also try to apply with a tiny brush a small (...A SMALL... ) amount of nitrocellulose thinner at the base of your piece. You shoudn't "paint" with the thinner because painting action can damage the plastic. You'll have just to put a little drop of thinner and wait its evaporation. Thank to capillarity, the thinner will enter into the joint and will weaken your bonding.
Some little damages can be fix with a coat of mr surfacer 500

Good luck