I don't know about replacement barrels for the No.1 Mk3, but I have a new Criterion barrel on my No.4 which is splendid and accurate.
http://criterionbarrels.com/lee-enfield-no4
I have .303 headspace gages in the shop, I will root around and find them.
If I am lucky in doing so, I'd be happy to send and loan them to you.
It is feasible to correct headspace on the No.1 by soldering a shim onto the boltface, although a correct replacement bolthead is preferred. I have an assortment of boltheads for the No.4, but none, unfortunately for the No.1.