No, FusionAuth doesn't support adding JWT headers to FusionAuth generated JWTs. I looked at the code and don't think it'd be a ton of work to add support; there's already some scaffolding in the fusionauth-jwt OSS project.

I highly encourage anyone with this problem to file a feature request here with more details about your needs:

We consult that in our roadmap planning. We also offer professional services if you need us to build it on a schedule. Please send a request to our sales department if that is an option you'd like to pursue.

An alternative would be to build a service that would re-sign your JWTs from FusionAuth with the needed header changes. Not optimal, I understand, but another avenue that might get you what you need.