I'm having an issue with some records where they'll slip on the platter, especially while cleaning or spinning up.  Would I be able to get a pad for the platter (like some other brands have) without messing up the weight calibration for the tone arm / cartridge?

I think you'd mess the weight calibration up less than you'd change the vertical tracking angle of the cartridge as the record height is higher. How much that would change things depends a lot on the cartridge, I expect the nude contact line would find it a bigger change than the eliptical stylus. Whatever you try should be very thin.

ONe issue with cleaning a disc on the platter is you also tend to push the suspension down too much in my experience. It takes a very light hand with a brush to make it work at all.