Would this treatment help on the black water tank

My wife washes dishes with Dawn. The problem we have is that it is super concentrated. I believe that soap suds "live" in my gray tank, and as such continually foul my sensors so that they give false full readings. I often think about creating a highly concentrated mix of TSP to add to about a 1/2 full gray tank BEFORE we leave on a long road trip to "wash" the tank. I've never gotten around to it however.