The case bulge is normal and depends on the diameter of the chamber and the diameter of the case. The majority of American made .303 British cases run on the small side on diameter and will show a bulge when fired.
Both .303 cartridges below were fired in the same Enfield rifle, the Prvi case is a larger diameter and has thicker case walls than the HXP case.
What worries me is the fact that you are leaving a mark on a piece of paper you put behind the case, military surplus ammunition sometimes has rims as thick as .063 and minimum headspace is .064. American made cases normally have a rim thickness of approximately .058 and this makes me think someone changed the bolt head and your headspace is set under minimum headspace.
I have set Enfield rifle headspace under minimum and as long as your shooting American made cases with thin rims you are OK. BUT some milsurp ammo may not chamber so keep that in mind.
Below a No.4 Mk.2 fitted with two bolt heads that I use as my main Enfield shooter.
Below from a 1946 Australian book on the Lithgow No.1 Mk.III Enfield rifle.
You can not convert the No.1 Mk.III action to .308 even with a new barrel, the action was not designed for modern high pressure cartridges.
I would leave the rifle alone if it has not been modified and is still 100% military configuration, and just shoot it the way it is.
You will be money ahead to buy a Savage Axis or Stevens 200 and have a rifle made from modern steel that hasn't been tampered with or is half way worn out before you get it.