Not able to reload and needing a 6mm could be tricky. I have seen options for a 25-45 sharps. Finding a barrel would be the tricky part, but definitely not impossible.As mentioned, you'll be fine with any 223 based case in the Solo 300 upper. I've taken deer, bears and hogs at well over 200 yards with the 223 and it works fine. Deer at over 300 and one bear at 350, no problems.
@Taudisio does really well shooting his 6x45 out past 500 yards and I have no doubt he could easily take all of the above at that range.
If you're handloading, another point to consider: If running 223 cases at 5.56 pressures, especially with a windowed magazine, they make up for some of the lesser case capacity compared to the ARC cases. Those run at considerably lower pressures in an AR platform, although those cases definitely handle the longer bullets much more easily at typical AR mag length.
Edit to add

Ammunition

Also adding that x-caliber barrels has a 25-45 sharps reamer.