Point of curiosity: are the No. 5 Jungle Carbines the only Lee-Enfields to feature the manufacturer/date information on the left-hand side of the wrist-band?
No, most No 1 rifles carry it on the right side of the wrist.
No 4 rifles carry manufacture dates and refurb dates on the left flat of the receiver.
The black paint is a milspec paint called Suncorite, which was used by the Brits back then. It would be difficult to replicate. Either finish would be considered authentic for the period...
Good on you!! The sling should be easy, it's the same as a No1 or No4 sling and are not uncommon, the carry case may be a challenge.