I imagine 3mm will be fairly noticeable. I use adhesive labels and put them on base cards. I prefer the look and feel of clear labels over blank cards, but it's not exactly cheap to do this en masse.
I have had Staples print a few cards on 12point matte business card paper. This seems to be the best 'look and feel'. But they have to adjustment of their printer, but it has been a pain dealing with them to get them aligned correctly (I have 6 card fronts and 6 card backs (one of which is the randomizer, and the other randomizer is a placeholder in the sleeve), and 10 regular cards.
I have to trim them from their sheet and round the corners. They work OK.
I am hoping to hear from others if there has been any success on getting fan-cards printed (and/or cut/trimmed) at an affordable cost.