There's quite a difference to the calibres your looking at so I'll give you my take on it as I use/used all three.
.243
Great calibre which I've used a fair bot on both Sika & Fallow but haven't shot any reds with it. I even have a mate who uses the calibre for ALL of his hunting through NZ which includes Thar hunting. He simply uses Barnes projectiles in his and said the killing power is even better just with the Barnes projectiles alone. I never tried Barnes projectiles in the .243, only in my .223 which I couldn't get them to group in but I always used Targex 95 gr projectiles which are made her in NZ and proved to be lethal and very effective on the game I shot them with all the way out to 230 mtrs.
People also swear by the Federal Blue box .243 which I've never used & "Federal Fusion" I hear is also good ammo from all accounts.
Even one well know hunter who ran the water taxi in Fiordland who's name I can't recollect told me he just used his trusty .243 for Wapiti and Reds.
7mm Rem Mag
"Rules with authority" which was what an old American friend of mine once told me and sure enough it does but it also comes at a bit of a price in form of cost. One thing I don't know is whether or not you reload and this has an impact on you and your budget. A mate of mine has a stunning 7mm Rem Mag rifle but he's paying just under $100 per packet of 20 cartridges and if your going to shoot long range then you need to practice.
Reloading for any calibre makes it far more affordable but also enjoyable especially being able to tailor loads to your rifle and your shooting.
Because it's a big calibre it's also going to recoil more so I'm not sure if your recoil sensitive. I'm a bit of a sook with recoil so I ran 150 gr projectiles in my magnum and later played with heavier 162 projectiles but also added a muzzle break to tame recoil where it felt like a .243 or less but you need ear protection and other people who you shoot with will just hate that rifle due to the muzzle brake! You could also put on a suppressor as another option.
Because of recoil prior to muzzle brake I have a nice scar on my nose from the scope due to shooting the rifle at an angle upwards at some Thar in a set of bluffs. It hurt but also reminded me to take more care while shooting a larger calibre.
Because it's a magnum typically the barrels are 24-26 inches long so their not always the nicest dragging through the bush.
7mm-08
Great round where I have a nice Kimber Montana in 7mm-08 and also load for two mates 7mm-08's where we were using 140 gr projectiles but dropped drown to 120 gr Sierra projectiles which are flatter shooting and have been used to shoot some big bodied red stags, mate took a nice 14 point red last year that was just a monster and a few months later he was chasing Thar shooting two nice bulls.
Great thing about the 7mm-08 is that it will certainly take anything we have here in NZ and the ammunition is cheaper to buy than say the magnum ammo.
You can also cut the barrel down if you like to the "Bush Pig" 16 inches with over barrel suppressor or as my 2 mates have done, they have theirs at 18 inches making their rifles portable for both the bush and open country.
A few years ago I bought a Tikka T3 in 7mm-08 and as another member on another thread put it, it underwent a sex change to a .284 Winchester which is roughly in between the 7mm Mag & 7mm-08 and I've found that a great calibre for both compactness as it sports a 16 inch barrel with over barrel suppressor.
Recoil is fine even my 13 year old uses it. I've used it mostly bush hunting sika and shooting some big Reds out the back of National Park out to 330 mtrs but since moving south I've shot a bull thar at 400 mtrs and the other week a nice red hind at 345 mtrs.
Anyway... good luck and hot barrels whatever you may do.
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