Not a T/C lock but I did brown a case hardened GPR lock. I used LMF and it turned out pretty good IMO. I don't know how the browning will take on an engraved lock though.
Naval jelly will usually take the case colors off and prep the metal for browning by giving a light etch. The brown works best on a less than polished surface. Be sure to thoroughly wash the Naval jelly off before you brown.
I'd strip the bluing off. Best thing to do is take the lock apart. (Take a pic or draw a sketch so you know how to put it back together) Get a glass jar or bowl and soak the lock plate, hammer and hammer screw in a strong toilet bowl cleaner. ( I like to use sno-bowl) Use gloves and an old stiff toothbrush in well ventilated area. You can strip it clean as a whistle in less than 5 minutes. Rinse it well in hot water and dry. Now your ready to brown it.
I browned one with Birchwood Casey Plum Brown, and one with cold brown. They both behaved like you would expect, no surprises. BTW, the cold browned lock came out very nice. I have a pic, just can't seem to make my photobucket account work right now. Bill