Trying to turn a vintage wash basin into a vessel sink. How can I get rid of this standing water?
If you don't object to a transparent substance in the bottom of the basin, you can use acrylic casting resin, aiming for a product which specifically advertises itself as clear, as some of the available casting resins are not.

Mixing the product per directions and pouring it into the basin with a bowl under the drain will allow you to place the exact amount, while the excess runs into the bowl. Ensure that the basin is perfectly level prior to the pour for what I hope is an obvious reason.

It's typical to subject casting resin to a pressure pot to remove bubbles, but that's not necessarily practical in this situation. A heat gun (preferred) or hair dryer will help to clear bubbles.

As a possible bonus, you can mix a small amount of glitter or similar decoration if you wish to add a different touch, although that's not consistent with the flower motif currently in place.

This may not be suited if you intend to use abrasive cleansers on the surface, but they can be polished out with appropriate abrasives and plenty of labor.