Give it a good wash with mineral spirits to take off any excess oil and finish it with anything you want. I think something like danish oil might be good since it should mix in well with the other oil and there may be less adhesion issues. Is the client asking for polyurethane or something in particular?
Some water based finishes turn milky color if you put something hot on it like a coffee cup. I don’t think mineral oil does that.
I still think your best bet is a good mineral spirit wash, light sand to knock any raised grain down, then oil varnish, which still mix in with the other oil (you’re not getting rid of all of the mineral oil without planing the surface quite a bit, I think) and the varnish will give it a seal coat. Then if you want even more protection, a light coat of wipe on poly. That should stand up pretty good to regular kitchen use. You have to tell them that it’s not stone, and even natural stone can stain if abused. Some common sense goes a long way.
My son in law used floor poly on his Maple countertops. He steel wooled each coat so it didn’t look “dipped in plastic”. The area around the sink was still nice after 5 years when they sold the house and they didn’t have a dishwasher so lots of splashed water.