I don't think a bore sighter would help you a whole lot. It'll give you a ballpark idea of where the scope is looking in relation to the bore, but that's sighting, not accuracy. If the rifle shoots small groups some place other than where you're aiming- that's the scope. If it flings shots all over the place- that's another story.
Do you still have the iron sights on the rifle? You can verify with those. Are the bolts which hold the action to the stock good and equally tight?
Is there any way to try a different scope? And speaking of scope, I assume you've double checked all the mounting hardware and screws?
And, is there a chance you were thinking of a bore SCOPE? One of those would let you see the inside of the barrel, but really tells the average Joe nothing if he does not know what he's looking for.
One option to see if the stock has warped or is somehow affecting the barrel- Cut a piece of hard plastic about the thickness of a credit card into a 3/8"x3/4" , pull the stock off, and place the plastic in front of the recoil lug area where the receiver flat rests flat on the stock. Now tighten everything down. If the groups come together, you'll know the rifle will benefit from either bedding or free-floating (or both if you choose). If it does not help, then I would think it's the barrel. There's a cheaper rout to try on that that is overlooked a lot these days. Take the rifle to a gunsmith and have him counter-bore the muzzle back 3/4". What this does is remove any bad things, errosion, or wear that may have happened to or developed to the crown or muzzle. It'll be cheaper than a new barrel. And it adds new life to old mil-surps that have seen a lot of use and/or abuse. I was told that it was a pretty common practice back in the day before 1903, Garand, and M1 Carbine replacement barrels were readily available.
Rednecks... Keeping the woods critter-free since March 2, 1836. (TX Independence Day)
I'm going to use the words "clip" and "Long Colt" every chance I get. It grinds my gears to see new members attacked when we all know dang good and well what's being refered to.