It's always gonna be a minor problem, but there are some things you can do to lessen it. Firmer your lube, the better it stays in one place. Simply chilling it before you start cutting helps. The tighter the fit of your cookie cutter, the less lube gets moved around. You can clean your cookie cutter as you cut to minimize lube transfer. Being careful not to have your lube so deep as to get any on the nose helps. Lube there can not only come off in the seater, it can be transfered for other bullets before they get to the seater.
I don't recall seeing what lube you are using, but the addition of paraffin will firm most up and reduce tackiness.