Hmmmm I have been thinking about this some more and wonder with your shoulder injury why you really want a Lightweight rifle as it will of course increase the felt recoil. Of course recoil can be mitigated by the use of one of the new super recoil pads and perhaps coupled with usign a Past Recoil shield
but allt his might be averted by having a rifle of normal weight.
Now as I said it got me thinking so I got thescales out and out a couple of lightweight rifles on them so see what they weigh all ready to go hunting. As I said I don't have any Tikka's so had to use what I do have to hand
#1 is a BSA Majestic Featherweight of 1959 vintage in .270 so it's a long action one. fitted with an alloy one piece Parker-Hale mount holding a 6x42 scope and fitted with a leather "Cobra" type sling. It weighs in at 7lbs 12oz, the rifle weighs 6lbs 7oz with just the sling fitted.
#2 is a BSA Majestic Featherweight from the early 1960's in .243 so it's a medium length action and it's fitted with a Khales Helia 4L2 alloy tubed scope in alloy P-H rings and weighs 6lbs 14 oz but it has not sling attached and with the scope and mounts it weights 6lbs 3oz. This one has a recoil reducing muzzle brake machined into the muzzle under the foresight ramp of BSA's design it's called a BESA.
Both have the old standard BSA Vented recoil pad.
What I am getting at hear is that the quest for lightweight hunting rifles is nothing new but the issue of increased recoil is still the worry for a lot of shooters using them. It was this recoil issue which led to the development of the BESA recoils reducer back in 1955.
The felt recoil on the .270 is sharper, not really heavier, than that of another 270 BSA of standard weight I also have although saying that I just weighed it and it weighs 8lbs 1oz with the steel tubed Pecar 3-7x36 in alloy P-H rings.
The point about less recoil with the 25-06 is not really valid IMHO as I now have a rifle in 25-06 yet cannot really tell that it has lighter recoil and that rifle is far heavier weighing around 9lbs scoped. The recoil may not be quite as quick as with the lighter rifles in .270 but it doesn't seem to thump the shoulder any less
The 243 Winchester is an effective cartridge but some still claim it's a triffle light for your often larger US Deer species such as Whitetail deer. May I suggest you reserch the availabily of perhaps the .257 Roberts
I would also heartily agree with looking seriously at the 6.5x55 Swedish cartridge. It's a lovely round to shoot and is very effective on Deer to which I can attest personally. Another one I would also endorse is the old 7mm Mauser (7x57) it's track record speaks for itself and it's not agressive on the shoulder
.
Plenty of options