I happen to have the same mower and the same problem. The rear of the transmission gets dirt trapped inside and has to be thoroughly cleaned. It's a known problem and the transmission manufacturer sells a "debris" kit which is really a shield to cover the rear of the tranny. I purchased and installed the debris kit but found it made very little difference and I still had to (completely) drop the transmission (I believe you are correct that the gas tank is nearly impossible to remove) and blow out the debris. This happened so often I've pretty much given up on using reverse, and just go forward. I mow 3 acres and it's rough on mowers no matter how much picking up I do, because some areas are dirt, and some are weeds around my pond. Let me know if you want me to find the debris kit name. I have it on my computer somewhere.
Have a great day!
Pete