I did slug the barrel and got .355 on the button, give or take .0001 or so. A .356 bullet should be fine, shouldn't it?
Not if it's "give .0001"
What kind of gun is it? The lead I shoot through my CZs is .358" - a lot bigger than .356". Also keep in mind that commercial lead bullets often have a harder lube than desirable for handgun applications. The tumble lube suggestion by the other poster (floydster) might actually give you some improvement. Also where is the leading occurring in your barrel? More at the muzzle end or more toward the chamber?
You shouldn't have to put up with leading at all when using lead bullets. I can literally shoot thousands through my barrel without cleaning it, and I still just use a patch when I want it to look extra pretty.