I measured some of my CCI#41 cups (Lot L05P). The cup height is .118". It is not easy to make the measurement because the anvil support legs extend beyond the cup rim** and get in the way. The MilSpec drawing for the case shows the primer pocket depth .118 +.004 / -.000.
SAAMI recommends .003 to .008 below case head face for the primer, tolerance stack ups considered. The #41 cups are domed somewhat. I have found if you use "On Press" insertion it is easy to flatten the dome of the cup with the ram force. You reported .012 depth which is perhaps a little too deep.
Until you get to the "bottom" of the primer problem, I would recommend you suspend use of the Lee Primer Cup Cleaner, and just use the brush.
Also, see if you can modulate the force on the ram handle until you can get the depth back in the SAAMI limits.
Finally, check your as-fired shoulder-to-casehead face against your sized shoulder-to-casehead face and determine that your sized cases are no less than .003 of your as-fired. If you do not have a case gage to make that measurement you can easily make one, as suggested by AeroE.
Measurement = X - Y [ETA] **The support legs of the anvil are pushed back up into the cup upon proper insertion.