Repairing sardonyx stone

I’ve got a signet ring (9ct gold) with a sardonyx stone (blue on
black) that is rather badly damaged, and I’m trying to find the best
way of repairing it.

How difficult can it be to replicate the design?

After a lot of backwards and forwards with some UK based firms, I
have the impression that my best option is to make a CAD drawing of
the design and send them that, and they send me back the stone. Then
I go to a local Jeweller and have them remove the damaged stone and
set the new one.

(The design is simply two letters overlayed)

Any suggestions?