Have you thought about the 6.5 calibres?
What I am about to say might be a bit controversial and at the risk of being shot down in flames, I’m going to put it in a slightly different way.
As the intended purpose appears to be for deer shooting and you’ve mentioned Reds, Perhaps the question you should be asking is how well will the .243 stabilise and perform with non lead bullets compared to the .308.
My opinion is that the time is very quickly approaching when we’ll not be allowed to shoot lead bullets at deer of any species in the UK, let alone Reds in Scotland.
So, purely on that basis, I’d say go with the .308 ( not that I particularly like the chambering).
If you already have a .243 that you like and know how to shoot well, then I see no reason to change it. Far better to spend the saved money on a superb scope instead. An ability to shoot confidently with a gun is more important than the round. A .243 x 100 grain ballistic tip bullet travelling at over 2600fps drops reds as well as any other calibre if hit in the heart, head or neck. Bigger calibres just go through the deer thus wasting much of the energy. I always like to see the bullet expired on the inside of the far hide with perfect expansion. These animals dropped like a stone. Heart mushed. Speed is key, getting over 2,600fps makes a huge difference to the bullet expansion. You will still get this speed up to 200 yards with a .243. At the muzzle it is travelling at 3,100fps. In my opinion, this is the problem with slower, heavier rounds on deer. The chest area by the heart is actually quite narrow and heavier bullets just go straight through and waste their energy on the ground behind. Being quieter, especially with a silencer, you should get more chances with a .243 than a louder round such as a .308 or bigger. Just my experience, but you will be just fine with either calibre on the biggest reds if you shoot it straight. Of course the .243 is easier to shoot straight with less recoil, flatter trajectory, quicker recovery, better sight of the animals for a second target. I would not swap my .243 for a .308 for deer. If I was wanting to go after many long range deer then I would probably go for the .270 or 7mm Rem Magnum, rather than a .308. But there are very few long range shots that cannot be made shorter range by stalking a bit more, which in my opinion is most of the fun. In my experience, in a days stalking, for every long range deer you may occasionally pass up due to not having the long reach of a noisy .270, you will quite likely gain two or three much better chances from the low noise of a silenced .234 not disturbing the ground. With silenced .243 you can shoot a red on one side of a rocky hillock, then peek around the rock and see more deer 100 yards away unaware of anything. That never happens with a louder calibre. They have all gone miles and you have hours of hoofing to catch them.