The problem with carrying handles wasn't the removal or replacement of the assembly from the barrel but the removal of the wood handle part. It was the zillion TPI nut that was the problem. As apprentices we used the official stupid tool of course and it was easy to tip the nut and cause it to cross thread on the stem. As apprentices we were told that we could only commit then heinous cross threading crime once as thereafter, it was deemed negligence on our part and we would have to pay for the damage. It was a complete load of twaddle of course but it made you think twice when repeatedly stripping and assembling.
Out in the real world we just copied the carrying handle forked screwdriver from our elders which held the nut square! I seem to remember that the handle only came from Ordnance as the stem (27 or 30) and fittings as an assembly. But in case you needed any of the individual components, you always had a load of loose parts in the tray. The Mk1 wood handle part always got a complete battering so they were slightly turned down to Mk1 spec to use the smaller dia end plate which solved the problems
I understand that the complete carrying handle assemblies are readily available in the US