The .308 cartridge is one of the most versatile around, with a few factory loads using light bullet some what available.
Before getting complicated with rechambering, rebarrelling, etc, maybe try some factory commercially available loads like;
Hornady 125 gr TAP ammo,
Winchester 125 Gr PSP,
Remington 125 Gr "managed recoil"
or if you can find some, the discontinued Remington Accelerator SABOT .223 loads.
" In 1977 Remington introduced the Accelerator in 3 calibers: 30-30, 308, 30-06. These are a special loading of the standard cartridge, using a plastic sabot that enables a .22" diameter projectile to be discharged from a .30" diameter barrel.
Velocities of more than 4,000 feet per second are possible.
You can buy the sabots separately and load your own .223/ .308."
I've shot up a few boxes of the rem sabots, and a several hundred rounds of reloaded 123 Gr Lapua .310" bullets loaded in .308 Win cases, to duplicate 7.62X39 Russian ballistics. Both of these were reliable and accurate for me in my 14s,
however,
YPMMV
LAZ 1