There are lots of .243 and 7mm-08 out there, not so with the Roberts.
+1 So few it's not really a viable entry. I love it, but it scarcity on the used rifle rack severely limits your choices.
Without going into a ballistics program I am not comfortable on recommending a reduced load in the 7mm for 250 yards.
But, in my NSHO, the 7/08 is the only viable option... in your list. I have one in a Rem. M7 and it is one option for my granbabies. My youngest child is over 30!

So that leaves the .243. or 6mm Rem. You are going to find a much divided camp here in regards to that cartridge for deer.
Good coyote round at 250 yds...

deer? at 50 maybe...

Private joke... I love taking a poke at they 243 fans! They go BONKERS!!!

I would however throw three more cartridges in the basket for your consideration.
Good lad!

a. 260 Remington
This would be my choice.

Got one in a M7 youth. The action is to be the base for a .250 project; not to worry, I've got a M700 in 260!

I wanted to hunt with it this year and just selected a mid-level load since I didn't have time to work one up. 120 PSPCL @ 2660 fps. NO recoil!

I may have to keep this as the Granbabies gun!
b. 6.5X55 Mauser
Great caliber! I got 3 of them: military, custom sporting and a barrelled action (another project).
But your choice of used rifles may be limited.
c. 250 Savage( 250-3000) if you can find one
I want a 250 real bad! It's kinda a tie for first place between this and the .260.

BUT, the
"if you can find one" says it all.
The foregoing is just my opinion... And while my opinions are always correct, any opinion is kinda like a skunk's butt. It stinks to everybody but the skunk.
