Either will work just fine. The 6.8 will carry energy out a bit further but for 500 yards and closer, the 25-06 shoots flat. I can honestly say that most of my mule deer shots have occurred at 200 yards or less with a few that were farther.
If the guide is telling you the shot will be that close then it won’t matter, I have friends that make those shots with a bow. If there is any chance you’ll have long shots I would definitely go with the 6.8
I shot every mule deer with my 270 Winchester using a 150 gr Sierra game king at 2950 FPS . All one shot kills all pass through ,ranges from 150 to 400 yards. Both guns are good choices , pick one and go !
Given the closer ranges you will be bunting, I’d go with the rifle that you have the greatest confidence , and gets you on target the fastest. At 100 yards, there is a good chance that a smart, old Mulie(or another nearby deer) may be aware of your presence With the right bullet that will penetrate from an angle, either set-up will be effective on mule deer for more then twice the distance your quide has advised, and I would be prepared for that possibility. My personal preference would be the 6.8. Most all my mule deer taken with 6.5’s and 270’s from 100-700 yards.