Well, a bathroom faucet has a lower flow rate, so you don't want that in a kitchen. There are many bar faucets available that are designed for just this sort of situation, but I've seen plenty of people put in tiny bar sinks with huge commercial style faucets, too.

I would be concerned that a bathroom faucet would not provide enough reach. Our second sink is very small, but quite useful. Our faucet is 13" from the counter at the highest point of the arch, and 8" from the counter to the bottom of the faucet mouth. When turned on, it hits the drain bang on, with zero splashing (something I, too, was really worried about).

This faucet is by MOEN, and is meant to be a secondary sink faucet. We purchased the larger version of this same faucet for our main sink....which also hits the drain, bang on, with no splashing.

We purchased both from Amazon.ca (we're in Canada), but I also saw the same faucets on Amazon.com. We bought them 2 years ago.

I think that several companies make the bar sink version for their faucet. I know when I was looking at Delta Trinsic faucets on ebay, several came up as the smaller, bar sized version.

Ours is a Delta Trinsic bar/prep faucet with a 13 1/2" ht. No splashing problems.