I would suggest a 7mm rem mag or 7STW.
26" barrel for 7 rem mag, 26 or 28" for 7STW.
7mm mag try the 168 berger or 162 a-mazx
7STW try the 180 berger.
You could go to 7RUM but if you are getting a 338 edge it would be getting closer to its performance.
I have a 7mm rem mag with a 26" No 5 barrel. I use the 168 berger or 162 a-max at around 3080 fps. It has worked really well on deer size game out to 638 yds.
Hope this helps,
Thanks to all have replied. Looks like some fun research for me to do. Keep the suggestions coming.
7mm-300 wby best accuracy load with the 162 A-max was 3440 fps using 7828 powder.
I'd suggest something other than a 7mm Remington Mag. IMO, the performance gain of the R-7 is NOT that substantial unless going beyond book max.