I see, you are wiping off excess lube by spinning the bullet, and the check doesn't spin with the bullet.
When I wipe off excess lube, I slide the base across a paper towel.
I'm sure a fatter shank would grip tighter, the problem is, if it is too big, it becomes difficult to install the check without shaving lead or failing to seat squarely. A crooked check hurts accuracy, a slightly loose check does not.
A separate check-seating step can fix a crooked check, but it's an extra step, and on small bores and bore-riders, the force required to seat the check can unintentionally bump up the bullet nose.
When I make a mold for myself, I try to make the check fit on the loose side, for easy check installation, but when I make a mold for someone else, I aim for a tighter fit, because many people are worried about the check spinning on the shank, for reasons that have never been satisfactorily explained.
The check shank is subject to quite a bit of as-cast variation, so a perfect fit may be elusive. Again, I'd rather err on the side of easy installation, and as long as the checks don't fall off at the bench, I'm a happy guy. But that's just me.
A concave and/or hollow rod for the "I" die may help reduce lube buildup. Also a hard lube. I'm sure you already know all that, just throwing it out.
Good luck with your oversize shank.