Where are you putting the anti seize? It's not intended to be a lubricant for moving parts like bearings. It's intended for things like screw threads that you don't want to seize together due to galling when tightening or from heat and/or corrosion.

So using it on the lock nut threads is fine. But for your bearings, I'd stick to a good grease. But even if you do fill your BB with it, you likely won't hurt it, unless you can spin at a couple thousand rpm.