Ive been doing a bunch of research on this topic as well. The wife has done a few mountain hunts with me, and she needs a lighter rifle to carry. Currently using a weatherby vanguard 270 with syn stock, and its just to heavy. The rifle I'm going to mention is a bit over your price, but its the one Ive been eyeing up for a bit now. Even if its not the rifle for you, the caliber research might be beneficial. I personally feel that if Elk are an option for your wife to hunt, the 243 is a bit too light. Originally I was dead set on getting her a Kimber Mountain Ascent rifle in 6.5 creedmoor or 7mm-08. But then I saw the Kimber Adirondack rifle, which is a little shorter and a little bit cheaper too. Its also available in 7mm-08 and 6.5. From what I have read and in talking with different people, the key to these calibers (on larger game) is a good quality bullet, and keeping shots under 400 yards. So that's the direction I am heading at the moment.
Now if I can just get those funds together....LOL...sigh. Good luck in your search! I am curious to hear what others mention for different rifles as well, so I can go check them out. I wish the wife had never gotten to put that Mountain ascent rifle into her hands and shoulder it at cabelas last year....when she did that...she decided it was THE one. LOL
I have no experience with either of those calibers mentioned, I'm only going off the data that I have been looking at online for the last few months, so please bear that in mind.