Non professional idea # 1
Can you find a modern fitting that fits inside the lead (that fits with nowdays std fittings), then use a ‘space filler’ – like old plastic bags etc to force pack around the insert and then cap it off with a watertight seal (araldite, putty, mastic) then just use modern fittings to link.
ie will a piece of white pipe go a couple of inches down inside the lead and seal that then use std bits to link white pipes.
To clarify I’m suggesting you use packing material so that the ‘rigid/hard to get out/messy’ seal material is only a thin layer at the top of the join (ie then you can get it out when it fails/doesn’t work etc). Also if you get rigid sealant down inside the lead I suspect that might cause bigger problems.