There's lots of opinions on oil. But any of the oils recommended here are safe for your RX3 and worth a try. If you don't like one, drain it and try another.
Personally, I'm a big fan of synthetic 10w-40 Racing 4T motorcycle oil.
Even my wife noticed the difference between the Castrol and the AMSoil I put in her bike last time ... quieter, smoother shifting, better mpg. But it costs more, so there's that.
I've tried Rotella T and T6 in my Vstrom, but they seemed to gum up the shifting and I didn't like it. I drained them each early and then switched to Mobil1 4T. Much better. AMSoil was even smoother