You should be able to find a new No. 4 replacement barrel in .303 fairly inexpensively, but unless you have one of the better actions, such as a late WWII Cdn Long Branch or Savage I would not put much money into any gunsmithing, as these guns aren't worth much on the used market. The metallurgy on many UK WWII manufacture .303s is questionable.
I would stick with the original .303 caliber, as ammunition is common and cheap, and the rifle feeds more reliably and is safer in its original caliber. Even with the better late actions, it is still rear locking and is marginal at best when converted to 7.62mm NATO.
I have a custom No. 4 Mk2* Long Branch which was rebuilt into a half-stock, target rifle, resembling the Enfield Envoy, with quick detachable scope mount, and A.J. Parker 8/53 target conversion of the Mk.I battlesight, done about 25 years ago. Its hammer-forged barrel was made by Heym in W. Germany and has a heavy barrel of contour similar to a Remington 700 varmint, and has standard NATO military 7.62mm, 4 groove rifling dimensions, but it was chambered in .303 British. The gunsmith who did the work used a reamer made by JGS from a Winchester accuracy test barrel print dated 1940, used for acceptance of Lend-Lease amunition. You could just as well use a SAAMI dimensioned pressure-velocity barrel reamer, which has similar dimensions, and which will give much better case life and general accuracy than the typical sloppy "trench chamber" normally found.
Accuracy is excellent with any normal factory .303 ammunition, Mk.VII and Mk.VIIIz, as well as commercial softpoints. Action operation is smooth and rapid fire opening is easy with no pressure signs resulting from use of standard .303 lead-core ammunition in the tighter .30 cal. barrel. My rifle also loves handloads with common .308" jacketed match bullets and groups as well as an accurized M1 rifle back, about 1.5 moa out to 600 yards. Beyond 600 it will out-shoot all but the very best Garands, due to the strange "compensation" feature common to the No. 4 in which individual shots having slightly lower muzzle velocities are launched at a slightly higher angle of departure than higher velocity rounds, which tends to reduce vertical dispersion. This quirk is very well documented and produces interesting results beyond the "cross-over" point, which occurs at about 800-900 yards.
I couldn't afford to have this work duplicated today, but am sure glad that I found this particular one, which outshoots a mid-1950s Holland & Holland reworked No. 4 Mk.*T which it replaced. If you ever find a Commonwealth Target Rifle in .303 do jump onto the opportunity!