I don't know about the price of the conversion? My Baikal air pistol was incredibly inexpensive relative to its features and performance (about a tenth of what an equivalent performing German one costs now, though the German product would have better fit and finish). Check parts availabliity. It may be you can get a barrel set with the 30-06 chambering and ejector for less than either reworking the original or a Savage?
Most of the loading manuals use .308 bullets in all these calibers. If you look up the original metric rounds, some illustrations put the bullets at .311" and some at .309". Government spec for 30-06 and .308 barrels is something like .3065" to .3095", if I recall correctly?
Out of curiosity, I grabbed a couple of boxes of 7.62 x 39 from the basement. One is Winchester, the other Lapua. The bullets mic'd at 3.0980 and .3105, respectively. If you slug your bore and it matches, that is still small enough that a .308" will bump up into it. Some European shooters report very good accuracy with slightly oversize bores in full loads. The main thing is the barrel must be straight. Shooting cast bullets through a bore larger than they are will cause leading and gas cutting, so they would have to be sized larger. Also, any bullets you find in this diameter can be loaded in the 30-06 case without special tooling. I don't know how heavy you can buy them commonly? Lapua lists a 200 grain FMJBT that is .310.
Nick