Have you seen Staceyneil's post on her bathroom remodel from several years ago? It involved a chest of drawers made into a vanity with stone counter and an under mount sink. I will link it below. Otherwise I have no helpful information. Stacey might answer if you post on that thread, but she isn't around much anymore.

Here is a link that might be useful: DIY budget elegant bathroom...

I can't help you with a way to convert an undermount sink to drop-in. I think if you want to keep the sink, enduring's suggestion of adding a stone counter and undermounting the sink to the stone makes the most sense. And Staceyneil did a great job in her redo, too!
It is possible to undermount the sink to the wood top:

[Eclectic Bathroom[(https://www.houzz.com/photos/eclectic-bathroom-ideas-phbr1-bp~t_712~s_2104) by Duxbury Design-Build Firms ARCHIA HOMES

[Eclectic Bathroom[(https://www.houzz.com/photos/eclectic-bathroom-ideas-phbr1-bp~t_712~s_2104) by Santa Barbara Architects & Designers J. Grant Design Studio

In the comments from the first picture, the designer commented that the paint finish lasted about 3 years, after which it was replaced by stone.

Thanks, jrueter -- I hadn't thought of using the wood as the overmount material (duh) -- I really like the walnut counter breezy used for her kitchen so I *think* it's possible to finish a wood countertop adequately to deal with the water. I must say though that when I tried it a couple years ago I failed miserably. It's rotted really badly now!

I remember Stacyneil's great stuff! I love her kitchen as well! Thanks for the reminder, enduring.

I think I just need to find a really really big stone yard with scraps all over the place. Last time around we used a pretty small yard and they were a pain. Plus they -- oh, I don't want to go into it. Painful... :( A neighbor is a woodworker and he is so down on trying to get wood sealed well-enough to be a countertop with water on it. I understand his point, so at least should just attempt to find a reasonable-enough stone. I prefer the wood aesthetically, but.... I'm surprised though that there's no commercial solution, i.e., special tile for modifying an undercounter sink to drop-in.

Have you thought about Marine varnish? The stuff they use on boats. Many coats on ALL the edges too. Why wouldn't that work? OR the 1/8" thick glop used on bar tops?

-Babka

Have you thought of soapstone remnants? I made this sink but you wouldn't have to go that far. The stone, if a softer variety, is soooo easy to work with. I did this sink, backsplash, and counter. My carpenter/cabinet maker built the cabinet to my concept plan. He did a wonderful job. The soapstone I used was mostly 13" wide stone in longish strips, left over from a real job. The bottom of the sink was a larger piece so the bottom of my 18" square sink has a "one piece" slab. The counter was pieced together. The backsplash section was epoxied onto the boxed sink. I paid...wait for it...$5/sf for these scraps. But you have to be willing to do the making, and the discovery of everything soapstone :)

Hope I am not boring you with my frequently posted picture.

I love soapstone. I put it in my kitchen 8 years ago and still smile.

-Babka