There is a traditional UK signwriter named Ron Lloyd, who does outstanding painted bar tops like you would not believe.
Could you not take a page from his notebook and varnish over the prints?
(I don't know anything about prints)
The bar that we go to sometimes has those business card things embedded in a clearcoat table top somehow. Not certain of the process.
We just used the Outdoor deck grade Spray Can urethane and did 4 coats. They also have roll and paint on types, and many different grades of urethane or epoxy... We got ours at a lumber store, none of our sign supply companies carried anything we could use for our application.