An interesting thought, but the only reason I can imagine for doing so would be to reduce the mass coming back on landing. It would be like an airliner aborting to return to its departing point. They dump fuel to reduce weight and make the landing safe.
It might just be simpler to manage the logistics of waiting for enough launch customers who can share the same launch configuration. That way, you're always launching with a full tank and not going through all that work to set up a launch for an incomplete manifest. The fuel is apparently not the critical factor.
And speaking of full tanks, here's a couple videos showing the inside of a rocket fuel tank as it delivers its fuel. I always wondered what went on in there and now I know.
Falcon 9 second stage LOX tank. No narration. The strange movement at the end is when the engine stops and the stage is at 0g. Although the video doesn't show it, the fuel eventually just breaks up into blobs that float around in the tank.
Saturn 1 (not 1B) kerosene tank. Narration.