Either of those bullets are fine if-
Neck shot ot white spot. No off angle shots, and no shots you're not 100% confident in.
The big key will be proper bullet selection in an accurate load. I this instance I'd rank bullet selection as priority one, accuracy as number 2, with velocity a distant 3rd in importance. I also reccomend keeping the shots in the short to medium ranges, as placement will be fairly critical.
The 243 Win is fine for mulies in capable hands. My mother, grandmother, father and a few of my aunts have taken a few deer with a 243, but they had the patience to wait for the right shot.