My 2 cents...I don't tow with full tanks ever. The water slosh is like a 250-300# weight pendulum swinging around under your RV and there's no dampener baffles in the tank (usually). I've never really needed to have full fresh water other than one time and I filled up the tank at a convienience store within a few miles of the camp site. Unless you have a bunch of spare towing capacity and don't mind burning extra fuel, I'd prefer to keep things as light as possible when towing. Seriously, if you are towing in mountainous terrain with an extra 250-300#s, your TV will feel more strain even if you don't. If taste is that important, there are good carbon water filters that you can install on the kitchen sink that will solve most of that and they only weigh maybe 2 lbs.
Bart and Tina
2001 Chevy 2500HD 6.0 Silverado 4x4-Banks Power
1997 Jayco Eagle 5er 285BHS-flipped axles