The 243 would work just fine. I myself shoot a 7mm RM and wouldn't trade it for 10 243's. But I've shot 100's of rounds out of it and I am very comfortable with it. Shoot em both and see what you shoot best and that's what you should take.
Another vote for .243 Win for Deer and Pronghorn It's a great cartridge for those.
As others have said, if you plan on hunting anything bigger, you may consider something like a .270 win., 7mm Rem Mag, .30-06 Springfield, etc.
I'm one of those in favor of having one rifle and caliber for most hunting. While no one rifle and cartridge are ideal for every and all hunting situation, there are many that cover most big game hunting, i.e. deer, pronghorn, caribu, elk, moose, black bear.
In big game I only hunt deer, cow elk, and every few years - pronghorn. For these I use one rifle, my Winchester Model 70 Featherweight chambered in .270 WCF topped with a 2-7x33mm Leupold scope set at 5X. It's a light handy rifle that gets the job done. I reload my own rounds using 130 gr Speer Grand Slam bullets at a velocity of average 3050 ft/sec at the muzzle. Some may say that this caliber and bullet weight is not nearly enough for elk. Well I completely disagree, and every time I've seen a cow elk and had the shot, the .270 in 130gr has never failed to perform for a humane kill, ever! I know exactly how that gun shoots and exactly where my shots will hit at various distances, angles, temperatures, and altitudes. That's the advantage of knowing your rifle
.243 is great...just remember to practice. all those calibers mentioned above are great, but they're not worth a nickel if you can't shoot accurately. the animals will only end up dead, there is no more dead or less dead.
practice and you'll be fine.
The 243 and the 7mag are enough gun for all of your big game hunting in the lower 48.
Load the 243 with a good 90 or 100gr bullet and the 7mag with a good 150 or 160gr bullet and get both to shoot well out to 350 yds.
The only problem is you only have two rifles. A guy can't get by with only two rifles. He needs one for every hunting scenario.
If you're lucky enough to be on private land and can leave your stand up during the off season you can tie an old piece of cloth or something of the sort to your stand. I like to use an old camo bandana that I have. That way the deer get used to the movement and eventually don't bother even looking up when there is slight movement. This may just give you the time to draw your bow or raise your gun without being detected.