I believe IKEA has a 8" round SS sink. Don't know the price.

I'd be careful about getting too small a sink. With beverage service, it is very easy to see rinsing a glass, washing off a lemon, washing hands, etc. and either of those sizes will make it hard not to be splashing. If you only have a water tap, you might be able to get away with an inset drain and reduce the temptation to use the spot for other functions. Or concede and get a sink that serves many functions and uses you are not anticipating.

We have a coffee bar sink, used for filling water pitchers, the RO spigot, and other water-only things. it's 15x15 and it's big enough to hold the teakettle under the faucet and fill it.

Much smaller and water would splash all over, and I wouldn't be able to sit the coffee pot under the RO tap.

A 6x6 sink is used with a push-spigot in a restaurant for filling water glasses, but it's useless for handwashing or anything else.

The costs of installing a small sink or a larger sink will be virtually the same , might as well put one in that will serve traditional sink functions -- if you need the counter space just get one with a cutting board made to fit over it when not in use. A few people around here have done that when space was tight and got great results.

12x12 is as small as I'd ever recommend for a bar sink. And that is tiny for a sink.