You can do that with a lot less than what is currently being discussed above. Something in 6.5 or 7mm is more than enough to do it nicely. A 300 of any flavor would the max needed for this task if you just want to burn powder. JMHO
Except those 6.5 and 7 or even the 300 are not allowed in the OP's area.
I see, I over looked the greater than .357 clause in the original post my apologies.
Given that detail
Gaschekt’s 200 TTSX in the Whalen sounds like it could be a good combo of speed and weight. There should be plenty of energy left at range for the TTSX to expand and do it’s job well. I know the op said no super hot rods but the 358 STA would have some cool factor here too with that 200 grainer in a 14 twist at 3250 or down load to 2950 and keep the 12 twist.