It's certainly possible to run a non-pumped supply, the difficulty will be down to a few factors such as location of pump and nature of the plumbing from the pump to all outlets in the bathroom. If the pump is remote and just feeds a single hot and cold supply to the bathroom, you would need to run new pipes to supply non-pumped feeds to the sink. How tricky that would be would be down to location, distance to cover and what walls, floors, etc. the pipes would have to be routed through.
What was the water pressure at the taps like before you put in the pump? If it was a trickle, then it'll return to a trickle without the pump.