TubeNet

I have never seen a fingering chart, but it is easy enough to work out if you have played a 5 valve CC tuba.

Same fingerings as a 4 valve Eb down to A below the staff, then:

Ab/G# = 5+4
G = 5+1+4 or 2+3+4 (both work on my 2040/5 - I use whichever feels easiest)
Gb/F# = 5+3+4
F = 5+1+3+4
E = 5+1+2+3+4

This is a really an interesting and difficult question to answer. First, I use the long 5th valve tube, so I have the 5th set up as a 2-3 substitution on my Willson 3400s. I get A = 4-2, Ab = 4-1, G = 2-3-4, F# = 1-3-4, F = 1-2-4-5, E = 2-3-4-5, Eb = 1-2-3-4-5, and, after those fingerings, you are into pedal tones, which in my experience are pretty well in tune and, except for the pedal C which is more nearly in tune with 3, alone, than 1-2. But, your results will be different if you use 5 as a flat 1 and your results will not be anything like any other 5 valve bass tuba, especially if they are using a 5 = flat 1. I have checked all of what I am telling you with a Korg tuner and the fingerings I have given you work very accurately for me. Further, B natural below the staff is much better played 5 alone. C, virtually anywhere on the horn, requires a first slide pull to center that tone. I am using a G & W Churada with a Euro shank. You need a Euro shank mouthpiece for this horn to make it work properly.

I hope this is a starting point for you to experiment and get what you want out of the horn. Congratulations on getting a really fine instrument.