Pardon me if I'm just missing something obvious, but what is wrong with "Lapislazuli"? I don't think I understand what you mean when you say it "doesn't deserve a consistent English pronunciation". Can it not just be read the same as you would pronounce the name of an ordinary Lapis-Lazuli - which is a real thing and does have a consistent English pronunciation already?
The pronunciation written out in simple terms is a bit like:
Lapis: L as in lie, short a as in bad, p as in pie, short i as in bid, s as in sigh
Lazuli: L as in lie, short a as in bad, z as in zoom, long u as in cute, l as in lie, i as in the 'y' in 'happy'
Alternatively, the final i is less commonly read as a long i as in tide.
I don't see anything about Lapislazuli in particular as a name that doesn't look consistent. Maybe I missed something, in that case, my apologies.