For elk, any of the Barnes TTSX will work fine at ranges out to 300 yards or so.
You could run the 110 grain and do just fine, but the sweet spot is the 130TTSX over 53/54 grains of Varget.
130 TTSX...that's the route I would go for an Elk load.
You can drive it fast out of an '06 and it will still be very easy on the shoulder. Some pretty big guys I know like this combo as an all-around load, because it's a knock-down laser inside 400 yards.
Another good choice is the 125 Nosler BT...it's a lot tougher bullet than it might appear at first glance to those who have never tried it.
I didn't know any better back then, but the '06 was my first rifle. I bought it used at 12yo. I guess I was a tough kid at 12 but so were my friends, and we all shot '06 and 8mm mil surplus loads just fine. 30-30s actually kicked a lot more from the bench.
Factory 180gr was no problem. We shot Win Super-X when we could afford it for the brass, but my reloads with Hornady 180 and 4064 kicked more. Since the RAR is a couple of pounds lighter than our rifles were, the 130 gr will tone it way down.
In a reduced load, I would use 4895, as it's more consistent than Varget at reduced pressure.
Actually, as I am remembering it now many years later, the blast and kick of the big bores was part of the thrill for us, and probably why I am still here!