Whomever said the .243 is a 6.5mm was a bit off. The .260 Rem is a 6.5mm, and is a fine, tempting round. The .243 Win is a 6mm, and is also a fine round. The 6mm Rem is a ballistic twin to the .243 Win, based on the x57 Mauser cartridge. It probably has a better case design from the reloader's perspective, but it isn't a commonly available chambering anymore. The .257 Roberts might actually have a bit to recommend it above either of the common 6mm cartridges, but it, along with most .25 calibers (besides the .25-06) kind of died off after the 6mm and .243 came along.
In that power class, in a new factory-built rifle, you've got some choices, but you're probably going to get a .243. There's nothing wrong with the .243! I got one this summer, and I love it.
Back to the long range precision and tactical matches, 6.5mm cartridges certainly do win them regularly, but so do 6mm cartridges. Honestly, though, is that really germane to this discussion? It takes a talented shooter to win one of those competitions, and while one rifle or caliber or chambering may give a shooter an edge, or a perceived one, if they were all limited to the same rifle and caliber, you'd see the same guys getting roughly the same scores and winning about the same amount of comps. Yeah, at that level, 1st, 2nd, and 3rd might change places because of one of those edges, but to a guy looking for a light-kicking, flat-shooting deer rifle, does that even matter? I can't imagine that it could. We're talking about guys hitting tiny things at 600-1000 meters, not deer in Florida.
OP is located in Pensacola. If he wants the lightest-kicking, flattest shooting caliber in the group he picked out, he needs a .243. It'll get you hits on stuff way past where most should be shooting at deer. Brass is common, bullets are common, and he'll have a ball with it, and his fiancee will probably like shooting it more than his .308. Everybody's got a different shoulder, but most recoil-sensitive people get along with a .243 better than a .308.
.223 is even lighter kicking, and has a pretty flat trajectory. It is enough of a cartridge, when loaded with sensible bullets, to cleanly take a Florida swamp deer. But OP said it was really not what he was looking for. There might be some bruiser of a whitetail in UP Michigan, or a mulie in Montana that it would be foolish to try to take with a .223, but again, that's not what he's looking to take (at least, basing my assumptions on his home location). If this rifle is really for the fiancee, then he might end up with one, but if it's to REPLACE the .308 (not that he said that it is, just something that crossed my mind), then he'll be happier with a .243, while letting her shoot something that is more pleasant than the .308 for smaller-statured folks.
OP, I wouldn't replace the .308, if that's what you're thinking - one rifle, one caliber is fine for one person, but if you're hunting together, you want two, anyway. If you're tired of the .308, that's another question altogether.
Anyway, I think you should get a .243. In case I didn't make that clear before.