Will it freeze

I don't know how these outdoor kitchens are plumbed. More than likely, there are no shut off valves going to the outside kitchen from the main lines going to the other parts of the camper. My trailer has a small outside kitchen (sink and stove) and the sink has no separate cut off valve. If yours has no cut off valves for the outside kitchen, then blowing lines out won't work, nor will pumping pink antifreeze. I suggest taking a little time and installing a cut off valve to the lines feeding to the outside kitchen with a low point drain. This way you can turn the water off to the kitchen, drain the lines, and still have water access to the rest of the camper.

The above posters are absolutely correct about letting your water trickle at the sink and your drain pipes freezing. Unlike a house, those drain pipes are under the foundation of the house and never freeze. In a camper, they are above ground and will freeze.