I have 2 old Eb tubas that I play often (a Giant and a small one) and the key with the intonation seems to be tuning the horn to the 2nd partial and using a few different fingerings if need be higher up. Also, buzzing the exact pitch like you owe it money.
I used an American shank Bach 18 in my old small shank small Conn Eb with absolutely no problems, a shallow no-name 30mm in my small (may actually be the medium) King Eb and a PT48 in my Conn Giant with very few problems (and they're there with different mouthpieces as well).
The few I've tried actually don't work that well in a bass trombone or euphonium, though a modern contrabass trombone mouthpiece does. I don't know why. My contrabone mp is actually deeper than the no-name.