The original poster was primarily concerned with mowing steep hillsides, not traction.
In my very cheap opinion, the R4 tires loaded will give MUCH more stability than loaded R1 tires. They hold more fluid and are wider. The OP would need new rims for R1's, big $...to me anyway.
I'd load the R4's with windshield wiper fluid, be sure the rims are at the widest position. Then try the slopes again. If you're mowing with a FEL on the machine, try taking it off and putting some low weight on the front. I've never seen wheel weights for R4 tires, but you may be able to make some with quick crete and some threaded rod.
Once you load the tires you'll notice a BIG difference in stability. I was amazed. My machine is about the same size as yours and I feel MUCH safer on it with loaded rear tires.