In factory loads, the .45. With suitable handloads, either.
But unless you're using a big Ruger SA, the thinner chamber walls in tthe .45 require the .44, if you're loading hot. Elmer Keith had discovered that by the 1920's!

I'm sure that your Skelton load will work to at least 60 yards, but is on the light side if you're deliberately hunting a deer, as opposed to taking a target of opportunity under ideal circumstances.

If you hunt with a handgun, I really think the .44 Magnum is a better choice, and you don't have to worry about acccidentally slipping a "Ruger only" .45 round into an old Colt New Service or SAA. Or, a M-25-5, this being the S&W forum...

T-Star