Thats exactly what I do. I have a blanket that I keep with me. Its pretty thin. I usually ask for a table against a wall or in a corner and then keep the blanket on the stroller but uncover the side towards us.
Thats looks like a neat stroller. I wish they had more pics posted of it though. I was trying to see how dark that mesh is. My mesh is pretty dark and conceals pretty well even without a blanket.
This is the one I have.
Amazon.com: Pet Gear AT3 All-Terrain Pet Stroller...