I recently got a brand new Schecter Blackjack SLS C-1 P and the action on it is pretty much perfect and it plays great. Only thing I want to change is to a set of Light Top/Heavy Bottom D'addarios (I think it's 11-52) for drop C.

What I want to know is if it'll be necessary to get it setup for this, mainly truss rod adjustment. I'm comfortable adjusting the bridge and fixing the intonation, however I'd rather not do anything with the truss rod if possible. Will the action remain (more or less) the same if I just change the strings and adjust the bridge?
Most likely you'll have to adjust the truss rod once you change tunings and change string gauge. Since the neck will bend in some way you'll have to tinker with the truss rod and adjust the action then accordingly. I too had initially some reluctance to touch the truss rod, but I recommend you read up on it and watch videos and give it a try, if you are into guitars.
LTHB is 10-52. 11-52 would be medium top heavy bottom.

Why is everyone so afraid of truss rod adjusting?!? You can turn an eighth to either direction and it won't break the neck. You're now using 10 to 46 in E STD? I would go for 11-56 if you want to compensate for the low C. And if you can, go for these.
I love NYXLs. They're that much better than the "normal" nickel roundwounds and don't cost that much more.. I mean, I don't cheap on guitars. Both of mine are worth well over 1000 euros each, why would I cheap on strings?
Gear pics

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If you are going from 9s in E to 11/52 in drop C the truss rod adjustment will be minimal as the guage change will primarily compensate for the drop in tuning. If you have a micrometer check the string height first at several points on the board, and don't turn the truss rod with the strings at full tension!