The OP said deer size game and keep recoil to a minimum. It follows that a high bc 6mm is mo bettah.
About 2 decades ago, I lusted for a 6-284, I was more about velocity then, as opposed to B.C.
I currently have a LA Model 70 Classic which I have wanted to use as a donor.
What would be the best 6mm chambering given that platform?
My first guess would be the 6 mm Remington AI - But I am wide open to recommendations.