Wet Bar Pump

I want to install a small wet bar sink in my den which is across the house from all the plumbing. I can get cold water there with no problem, but a waste pipe could be difficult - a long run and lots of framing in the way. Since it's just a bar sink, can I put a small sump under the sink and pump/exhaust via 1" tubing similar to how the water softener backflushes?? Would that be OK with code?

Thanks!!