Well, it depends; the core designs are good because in case of a sudden war or whatever, you can start pumping ships easy, instead of taking some 10-15 minutes to play around the shipyard.
But in a 1-on-1, custom vs core hmm...depends; the core designs are available when you get the relevant tech attatched to them, while custom ships you can made them as you wish and put what you want on it...
Also, the core designs seems to favour laser weaponry; I had Laser I researched and Stinger IV, and the core desings that were showing up were ALL armed with lasers instead of stingers.
The point defenses were up there, so core design armour was following my defence techs...but weapon wise they did not.
I would say core will win; if only because you can tailor them to the demands of whatever situation you're in.
I removed the "give designs to player< option turned off, frankly, my TAN Mars class corvettes are way better than those Battle Axes (even more FASTA with my TAN Mercury support fregate wiht Fleet Bubble III bwahaha in that fleet)
Twilight of the Amor 2.0something version I got btw.