The .257 Weatherby will do what you want it to do with no problems. The velocity is really not needed for the ranges you describe, but it is the flattest shooting factory cartridge you can buy unless the 30-378 shoots flatter, but that is a whole different level of cartridge. You'd be well advised on a 26" if you want to see what it will do. In a shorter barrel the advantage over a 25-06 is not too apparent. Make sure you use premium bullets with this round, if you get a close range target you will need a good bullet to ensure it doesn't blow up on contact. I shoot a .270 Weatherby, which is based on the same case with .02 more bullet diameter, might be another option for you to look at as it offers more bullet weight for larger game but doesn't seem to have much more recoil, it feels like a 30/06 or 7Mag in similar weight rifles. The .270 with a 130gr bullet will do anything the .257 with a 120 will do. Don't pour rounds through a hot barrel with this cartridge, you'll likely see your barrel life suffer quickly if you do this.
If your handloads can beat factory ammo by much you are probably loading to unsafe pressure levels when it comes to Weatherby ammo, not to be confused with Weatherby loadings by other cartridge mfgs.