Cartridges that I would consider for you in a 14-15" barrel would be the 6.5 Creedmoor using Hornady's ammo with the 120 or 140 A-Max bullet (Match or Superperformance-Probably Match), 260 Rem (Fed, Rem., and Black Hills all make ammo for it) or the 7-08 (Hornady, Federal, Rem, Win. also makes ammo for it).
The Creedmoor would be the mildest of all three and would fit your parameters.
I have killed elk and deer with the 260 and 7-08. The Creedmoor is a newer cartridge, but I wouldn't hesitate to use it for your set-up
A good muzzle brake
is also something that will cause your recoil to become basically nothing. Holland's Radial is my hands down favorite.
You will need to wear hearing protection when you hunt. You should anyway with any handgun or center-fire rifle.
Give me a call if you have questions about a gunsmith or cartridge choices.