Do you have a sump (ejector) in the basement? I ran a drain and vent pipe along the perimeter from our back bar to there.

Good idea. Yes, we have a sump pump, but it is across the basement, about 35’ away. I wonder if that’s too far to properly slope to drain?

Presuming only liquid will drain from your bar sink, a slope of 1/4” per foot should be sufficient, ideal is 1/2” per foot.

I believe it could work, that only requires 9” of drop from the top of a trap. Do some measuring of a grade along the wall (apply masking tape) to mark the slope.

Just to be clear, when you say "we have a sump pump", what purpose does it currently serve and where does it pump the liquid to? If it's pumping ground water from around the foundation out into the yard, it can't be used for a bar sink.

There are products meant for this type of situation. The basin sits on the basement floor with no need to dig a hole. Here is one example:

https://www.zoellerpumps.com/en-us/products/package-systems/drain-pump