What would the company need to do to comply with licensing? Release the studio, shared library, and server-side components as AGPL?

Perhaps not the server-side components, unless they directly link AGPL covered code there too. No one that I've seen seems to be asking for the server side parts, just the client parts so they can use the printers locally they way they want without hacking around.

Studio is fully release IIRC, it is that "external" part which is the issue.

The linkage between that and the client should be soft enough otherwise for the server side not to inherit the responsibilities of the client. Though if part of their claims attempting to defend their current position suggest that linkage is more direct, then they are taking aim at their own foot somewhat.