I'd suggest a new washer that goes on the bottom of the counter top. It would be large enough to cover the existing hole in the counter top and have a hole in the middle that is the correct size for the new faucet.
This may be a tough ask, so you may need to make one out of 3/8 or 1/2" plywood.
- Using a hole saw, cut a circle out of the piece of plywood large enough to fit securely over the hole in the counter top.
- Using a smaller diameter saw, just large enough for the new faucet to fit through, use the same pilot hole in your disk to guide the saw to make the new faucet's hole.
- Since this will be used in a damp location, you may want to use pressure treated lumber, or, you may want to seal the whole thing in epoxy to protect it from any potential leaks.
If, of course, you have metal working tools (or know someone who does), you could make this out of a sheet of much thinner metal. You could probably use a hole-saw designed for cutting metal to do this, but I'd suggest that you use a drill press as you'll need to cut slowly and keep it lubricated to avoid overheating the bit & the metal.
Place your new faucet through the hole in the counter top. Place the washer over the stem from below. Screw on the mounting nut that came with the faucet.
You may also want to cut a "washer" to use as a filler donut to go inside the hole in the counter top to prevent the faucet from wobbling. Repeat the steps above, but make the outer diameter just large enough to be a snug fit into the counter top hole. A little bit of silicone caulk between the donut and the counter top should prevent wiggle and water ingress, yet still be removable if a larger faucet stem needs to be installed in the future.
The drawback to this method is that the stem of the new faucet may not be long enough to go through the counter top and the new washer (especially one made of wood).