Sry - I do not understand why or how you could be having a problem with thicker than .010" patches.
The thicker patches do require a short starter, as noted. The small peg of the starter is placed on the ball, then
the side of the starter knob is smacked with the palm. That puts the ball and patch into the bore 1/4" or so.
All of the starters below have a short peg for 'starting' the patched ball.
Some guys simply smack the ball with the "top" of the starter knob, holding the shaft in the hand and using a pounding motion.
I got away from this method, going to the short stud system, as when smacking the ball with the top of the starter, the ball rarely started squarely.
When using the short peg, this happens pretty much automatically.