BX2380 & tires
Papadiver -
I've had all...R1's (ag/bar), R3's (turf) and R4's (Ind) on a steep lawn. I've always said my tractors need to be part mountain goat.
The R1 and R3 were on stout garden tractors, the R4's on my BX2360.
They will all do damage given certain circumstances.
Probably the best bets are what skeets and Hokie alluded to...planning your path/turns, mowing when dry, etc.
I have a Sugar Maple I need to mow around in a steeper part in a corner of the yard. Regardless of machine...sod gets ripped up around it. Just a fact I live with after 27 years.
Personally, I'd like to see some pictures of what you're mowing.
Slope doesn't show well in pictures, but it's worth the exercise.
These are parts of my yard. The mentioned small maple straight ahead of the tractor. I dig up around that, and have for years. Also showing pics of my front ditch. There are times I probably couldn't get out of it with R3's.