Most aircraft have an ACARS system installed these days. It’s basically SMS in the sky.
I’d declare a mayday, tell Center I need to hold over some point while “troubleshooting a problem”, and send a message to dispatch using the ACARS system. ACARS is discrete, it won’t broadcast in the open.
None of the 320’s I fly have satphones installed, but when I was flying heavy international freight all of my 747’s had satphones installed.