The amount of spatial expansion that would be experienced on even the scale of a large solar system (even beyond the orbit of Sol and Pluto) would be negligible at best, and essentially indistinguishable from the probability of error on our current calculations. That fact notwithstanding; you are trying to ignore gravity again. An orbit is a complex thing; a careful balancing act of inertial forces (velocity, kinetic energy, etc) and gravitational forces (gravitational attraction between two masses).
Think of it like a table with a fine silk tablecloth. The tablecloth represents the "fabric of the universe". In the center of the table, you have a plate. Tethered to that plate by an elastic band is a saucer. The saucer has a velocity in a straight line, but is held a relatively set distance from the plate by the elastic band, causing it to orbit the plate.
Now, the tablecloth is quickly yanked out from under the plate and saucer. Do they move? Sure, perhaps a tiny bit - but because of inertia and the connection, they find that equilibrium again against such minuscule changes. Same applies on a planetary scale - if the change in distance due to cosmic expansion was sufficient enough, sure - the objects would fall out of orbit (and we would all be dead). As it stands, it isn't.
After all, if gravity is strong enough to overcome cosmic expansion on the scale of distances between the Milky Way and Andromeda, why would it be unable to do so on the scale of something so much smaller as a solar system?