My guess is that the shorty valve is a 1/4 tone valve for intonation.
The one next to it is probably a 3/4 tone or flat halftone.
Basically, play the thing like a three valve F tuba but add the 1/4 tone valve to 123 so a low B or low F# is spot-on in tune. The 3/4 step valve could then be added to give a solid low Bb, probably the lowest useable note on this F tuba. Keep in mind that most early F tuba literature did not go any lower than this.
FWIW, I find the flat halftone (3/4) valve to be pretty useful on an F tuba. I have to wonder about the usefulness of the 1/4 tone valve; a flat wholestep or two step (aka mirafone) would have been a better choice - nothing a competent repairman couldn't plumb in. Add a traditional length 4th valve in the right hand and you have yourself a nice little F tuba.