240s should work, but you'll want a fairly tough bullet. In the 240 gr weight class, bullets like XTPs and Remington JHPs/JSPs expand too quickly, and I wouldn't trust them. (The Remington bullets, in particular, are bad, bad, bad. I have varmint bullets that are tougher.)
For Elk with a handgun, I prefer a wide flat nose bullet that can penetrate, over almost any kind of expanding bullet.
There many commercial cast bullets out there, that would work well (if you reload). One of them would probably be my choice.
If you don't want a cast bullet, the Nosler Partition (now the "Sporting Handgun - Revolver), Speer DeepCurl, and, surprisingly, the Sierra Tournament Master (220/250 gr) are good choices.
If you don't reload... I'd probably be looking at loads from Buffalo Bore, HSM, Cor-Bon, Double Tap, Grizzly, etc. Or, even this Federal load (haven't seen it before this year, but it's quite close to what I carry):
300 Grain CastCore Flat Point.
If you look around, you may notice that the majority of .44 Mag hunting ammunition is loaded with hard cast, wide flat nose bullets, in the 270-320 gr weight range. 240s can work, but most people prefer something a little heavier and trade expansion for the wide flat nose.
When carrying my .44 Mag (7.5" SBH) on Elk hunts, it's loaded with lead 310 gr WFN bullets (at a moderate hardness of ~BHN 12) at what should be 1,150-1,200 fps (not yet verified).
Do yourself a favor, though. Double check with the friend in Oregon, and make sure you'll actually be hunting in an area where most shots are at appropriate ranges for a handgun. It wouldn't be much fun to get out there with just your handguns, only to find out that the average shot is 200-400 yards in that area. (Don't count on the rut letting you bring them in close, either. The rut can be stopped by weather, and take a few weeks to restart ... after the season has ended.)
