Back flushing water lines?

There is no way to get to the faucet lines without taking off the whole panel that is screwed and caulked to the wall.  That is where I am afraid we might tear the wall paper.  We unscrewed the back of the toilet, and ran the pump, but only a dribble of water came out.

Is it possible to back flush the cold water lines in the bathroom?  The toilet and cold water faucets in there are barely dribbling and it looks like we would have to totally take things apart, risking tearing the walls taking out the caulking around the faucet panel. I am thinking of making sure the water pump is turned off and using the garden hose to try to shoot water backwards through the toilet and the cold water faucet.  I am assuming the water tank would have to be drained and left open to clear out the grit that the pump must be picking up. When I first turned on the pump and water faucet it shot out air and water for just a moment.  We dry camp and don't usually even carry water in the tank, but are getting it ready to sell now. I already cleaned grit out of the kitchen faucet and the hot water is running fine both places. Is there any downside to trying to back flush the cold water lines? Thanks for any advice!!!!