I used to work at a wilderness therapy program where everyone ate with sticks. Most people fashioned wooden spoons and/ or forks each week.
For me, it feels weird to use a mineral oil on a food related product, even if it's a "clean" one. "Food grade mineral oil" is a strange association of words, in my mind at least.
But I saw yesterday some vids about making cutting boards. They use the mineral oil too (and then a mineral oil - bee wax compound as a finish).
Counter top guys said to put a clear sealant around the edge of the sink. I think once that is done, the wood under it won’t take oil. Then again maybe it will soak along the grain. Will keep it as narrow as possible.

So I thought maybe oil it first and then seal, but the sealant will probably not stick to the oil. 🤔

Mark around the sink, lightly, with a pencil. Remove the sink. Use masking tape to cover to the pencil line from the sink edge. A small amount of oil may seep under the tape from capillary action, but shouldn't present much issue. Seal the sink in with crystal clear silicone.
If it's weak adhesive, it might be easy to remove and reinstall the sink.

Can you find out what kind of adhesive?