Go with the .44 Magnum. The SRH you mention is a fine, accurate handgun, and any Ruger .44 Mag. is a very good value. Although you can spend more money, absorb more recoil, and hear more noise with other options, nothing is really any better for deer and other mid-size game than a .44 Magnum.
Here are a pair of custom SRH .44's from my corner of the world:
