Can we get a photo of the mark on your bowl?
A couple coats of polymerized tung oil over the paint? Well, to me, that is very good news! To my thinking, the oil likely did not penetrate through the paint very much, and any oil after the first coat likely cured on the surface, albeit very thin because I'm guessing you oiled, waited, then wiped off. But, there is a chance you have a cured oil film under that Sharpie, not paint under that Sharpie. That cured oil is essentially plastic. Just very, very thin, at best.
I'm guessing Sharpie gave you the Amodex idea?
Google AI says Amodex,
"Amodex stain remover is a
lanolin-based soap formula, containing no solvents, bleaches, or harsh abrasives. It is designed to be gentle yet effective on a variety of stains, including ink, grease, and food, and is safe for use on fabrics, upholstery, and even skin."
Lanolin is sheep's wool grease/wax. Completely natural if it grows on a sheep.
Other ideas to try at your own risk-
Accidentally swipe a permanent marker on a beloved surface? Before you deem your surface as "ruined," check out these tips for ways to remove surface from any surface in your home.
www.thespruce.com
This article describes using acetone, and shows it on a wood plate, which I'm presuming has a surface finish on it.
Get some cotton balls and Q-Tips. I'd be tempted to lightly dab a Q-Tip into rubbing alcohol or acetone, lightly rub just on the marker for a second, then immediately blot (not wipe/rub) the surface with a cotton ball to remove the now wet ink, repeat as needed.
Good luck.