What exactly is the Mo5's problem? You have the potential for a great "sporter" right there? It may have a simple bedding problem which is easier to fix than you might think. Most Enfields are designed for a full length stock. Fitting a short sporter type involves center bedding, something most Enfield barrels hate. That may even be the problem with your Carbine!
If you still want to go with a military rifle then I'd suggest picking up a non boogered sporter & working with that. I'm not a purist collector or anything but if you're going to restock you might as well get a $125 "Bubba" than a $350 "original" & save the difference in price!
A No 4 will adapt better than a No1 (SMLE) the sights & barrel configuration lend better to a sporter than the earlier No1 MkIII(*) one with its spring tension bore bedding & open sight with the short sight radius. It is also easier to scope a No4 than a No1 (SMLE).
1: All matching serial numbers, bolt (on the rear of the bolt handle) action on the wrist band or left receiver flat, Magazine & forend.
2: No giant oversize chambers, L-E's are (in)famous for them & it'll shorten case life drastically.
3: Not a 2-grove bore, there's nothing wrong with 2-grove bores in general but many, not all, Enfields hate boat tail bullets. The 2-grove ones seem to be particularly sensitive to the problem.
4: Bore condition. Most of the barrels have fired a lot of cordite & corrosive ammo. Make sure yours was taken good care of by previous owners.
5: Bolt head number (on the outside edge of the locking tab that hangs to the right hand side in the bolt guide rail) Lower is better, be suspicious of high numbers as the body may be approaching its useful lifespan.
6: Muzzle condition. Nicks or out of round (a common problem with rifles cleaned with a pull through repeatedly) those will make it a poor quality shooter. Do a "bullet test" against a known good muzzle to check.
7: Internet rumors. Most are rubbish at best, stick to facts you can verify.