The rectifier tube doesn't affect the tone directly by itself since the signal isn't going through it. That said, it affects the tone by how well or poorly it converts AC to DC in the amp, and this is realized mainly as a feel difference when you play the amp up in volume where its clipping.
With bass playing, you usually want a tight response and a clean or slightly dirty sound that responds quickly, so a lot of sag from a tube rectifier isn't really the best thing in a bass amp. It sounded ok with the Amperex GZ34, but it sounded better with the Copper Cap. The main reason I went with the Copper Cap instead of a couple solid state diodes like the 1N4007 is that they don't emulate the voltage drop like the Copper Cap does. With solid state diodes, the plate voltage was around 580v, and with the Copper Cap it was around 560v which Worked better with the cap setup I had in the amp. If you're using the stock cap setup, anything over 525v is above the rated max voltage on the caps and isn't a good idea, so almost all of these old Sunns need to have the power supply setup modified and every little reduction with the voltage helps.
My Vox AC30 sounds pretty good with the Copper Cap in it, and sounded fine with the Chinese GZ34 that blew after 10 years, but with the Amperex it was overall just richer sounding. Since that amp is cathode biased, that could just be due to how the power tubes are now biased with the different rectifier in place. The Amperex that I used was the only one I had, which has the DD getters and brown base. I got it from my grandpa after he passed. He was a ham radio guy and had a lot of cool stuff.
If I was using an amp for guitar, I'd want a tube rectifier in it, or a Copper Cap as they sound great too, but I wouldn't spend the extra cash on a NOS rectifier unless the current quality production dries up as the JJ is just fine. Rectifier tubes last a long time as long as they don't get mechanically jarred often.