I'll go against the 243 as I think it's honestly a bit light on Reds. Will it kill them sure but not as well as something larger.
For a beginners gun low recoil is a good idea but making things simple and reducing the requirements when making a shot is ideal. You dont know what you dont know when you are a beginner and buying something foolproof is a huge benefit.
I'd put the Creedmoor above the 243 but even that's a bit light (my main rifles a creedmoor and it doesn't kill like a 7mm or 308 tend too).
I'd have to say for balance of killing performance, decent factory ammo availability, and suitable ammo etc I'd be looking at a 7mm08.
Most 7mm08 ammo on the market is suitable for red deer whereas plenty of factory 243 on the market is targeted towards light game and predator hunting in the US. 6.5 creedmoor has plenty of options and it would be my second choice as likely that the precision hunter and the 143 eld x will shoot well and is a decent load.
308 is also a fine choice but has more recoil than the above. Realistically any of the 308 based cases 243, 6.5 Creed, 7mm08, or 308 will work fine with the lower down options giving less forgiveness and less killing power but typically less recoil better ballistics whereas the latter's have better killing power at the cost of recoil (which is still pretty mild especially with a suppressor). For me the best balance probably lies between the 7mm08 and 6.5 creedmoor.
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