I hunt The Charlottes every year, pretty much any legal centerfire will work over there.
The deer are small, they aren’t flighty and normally they’re harvested at close range.
I’ve stayed with the ‘06 and used a 6.5x55 for several trips; both extremely capable in almost any hunt in Canada.
My partner has successfully used his 223 Mini14 numerous times.
I had a 6.5cm but never hunted with it; however on the bench I never saw any difference worth debating over several other 6.5’s...
So I sold it.
IF I was contemplating another 6.5 for hunting (and I’m not) I would be considering one of the Magnum chamberings (264WM, 26 Nossler, 6.5-300 Weatherby). Pros aren’t worth the cons for me but they do offer some benefits...
Of the two you posted for the purpose you stated the 243 is the clear choice in my opinion.
Short action, low recoil 6mm with the right bullet (non varmint) will do the job easily.
270 is just to close to the ‘06 in size, weight, recoil and ballistics (given the game and ranges).