My coop has a BMC replacement filter in a standard intake box. Would I make any gains by replacing the whole setup with a K&N induction kit?

you might notice a slight decrease in low end power and an increase in top end power by switching to a cone filter. It will be noisier at high revs too, if that is to your taste

if you have a front mount intercooler then a straight induction pipe and cone filter where the side mount intercooler used to be will give you the biggest gains, and reduce turbo spoolup time.