Hi Wjdski
The neoprene washer and the rubber gasket should not come in contact in any way. If I understand your issue and the strainer you are using the neoprene washer would go on top (under flange) and the rubber washer would go on the underside of the shower. However, I never use the neoprene washer... actually throw those away.
Instead, I would suggest using a mold-resistant silicone caulking under the flange of the strainer and then use the neoprene washer on the underside between the nut and the base of the shower.
There should also be an anti.- friction washer that will go between the nut and the neoprene washer (see image)... this keeps the neoprene washer from slipping out like yours did the last time.
I know your drain is brass, but the plastic drain shown should be like your drain and should be installed as I said...discard neoprene washer (not in picture I posted, 'cause it shouldn't be used!!)
Oh, and by the way, the nut should be tightened very tight!!
Post back... let me know your thought...
Mark