The 243 is fine for deer, even borderline for larger game with careful bullet choices and shot placement. It is actually quite popular for long range target shooting. Just for reference in a typical rifle 243 will have about 11 ft lbs recoil.
Rounds like 6.5 CM, 260, 7-08 and 308 are great deer rounds too, but are also suited for game larger than deer if you ever decide to hunt larger game. For someone who knows what they are doing I have no problem with shooting game at longer ranges. But everyone has to know their limits. For me 300-400 is the max, and even then only under perfect conditions. Any of those will kill most anything at 400 yards If you want to shoot targets beyond 600 yards the 6.5 CM is the clear winner. It is still in the game at a mile, and with factory rifles and factory loads. A 243 will do it too, but not with factory rifles and factory loads.
The 6.5 and 260 have recoil much closer to 243 at around 12 ft lbs, but can shoot heavier 120-140 gr bullets vs the 243 which is best with 85-100 gr bullets.
The 7-08 and 308 are suitable for deer and anything else in North America. They can shoot much heavier bullets typically in the 140-165 gr range, but 175-200 gr bullets are an option.
Most people find recoil tolerable, about 14 ft lbs for a 7-08 and about 15 for a 308. But for some that is just too much, especially from a light rifle.
It sounds like 243 is a good option for you.