So, right now the mute plays about 20¢ sharp, which bugs me a lot. It is easy to fix, though, but it is a nuisance. My other mute do not alter the pitch at all. Additionally, there is a dead zone starting around low Bb down to about low F where it is unstable and difficult to center.
You might want to lower the resonance frequencies of the mute by making the mute opening smaller in diameter, and longer. try to find a piece of plastic tubing (hardware shop plumbing tubes?) which fits inside the mute opening, and cork it on one end for tight fit. maybe some 15 cm long or start with 20 cm and shorten.
Making the opening smaller in diameter and longer (it really does not matter much if the tube is mostly inside the mute) pulls resonance frequencies down, and might correct your issue.
I have two pieces of tubing for my wooden straight, each one fitted for one of my Eb tubas, and was able to get decent intonation for both this way. This is a cheap experiment.
Rolling tones on a bass trombone DW straight and cup mute I did resolve by drilling three small holes of approx 5 mm diameter in the bottom plate of the mute. I never did try this on a tuba mute, though, so others might want to comment.