Look up the 224 McDonald. It's basically a 6mm Rem case that's given the Ackley treatment then necked to .22 caliber. It's in the 4800 fps range with 40 grains and with 30 grain Bergers goes over 5200 fps. It was designed by Charley McDonald who lived just a few minutes from me. Unfortunately...
A .243 seems like the perfect answer. With 80 grain bullets there’s almost no recoil, ammunition is cheap, it shoots flat and it’s available in a broad range of inexpensive rifles.
Shortening a barrel to reduce FPS with the purpose of reducing throat erosion will not work. Simply put, most erosion happens before the bullet gets anywhere near the end of the barrel.
In my opinion it’s pretty hard to beat a Tikka T3 Lite. They’re not expensive (under $800 bucks) they’re light and they’re quite accurate. Available in 7mm Rem mag which will fill your elk requirements. Leaves you a couple hundred extra bucks for a brake or some fine tuning.
Canadian biologists claim that an adult wolf needs 16 - 18 full sized deer or 8 elk per year to survive. It doesn’t take much math to figure out that it won’t take many wolves to decimate a large herd of elk or deer.
I found that the best scope I could find for my money was a Tac Vector Paragon. Really good German Schott brand glass and very reliable tracking for less than $500-$600.
With my 257 Wby I use the same technique.... zero at 330 yds which is 3.7" high at around 225 yds and 3.7" low at 400. That way it's just point and shoot.