I run my AC year round it drys the air out and is especially useful in unforgiving the windows. In fact on Hondas you turn the defroster on the AC lights up automatically.
But dude seriously the dimming isn't a feature. So it's an electrical issue. Thus you need to diagnose or start down list of electrical components main two are battery and alternator. Did you get same size battery, did the battery get properly registered? Then turn things off like AC radio heated seats etc... Best of luck sounds like its bugging you enough that I hope you find out what the gremlin is soon.