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Because of the interaction of the mouthpiece cup, throat and backbore, there is no good way to predict how a mouthpiece will or will not perform on a given tuba, so in the end, you'll just have to try one. If you like the diameter and the rim shape of your Schilke, then here is a mouthpiece chart to get you in the same ballpark with mouthpieces of similar dimensions:

http://www.ibowtie.com/tubampccharts.html

That said, take my experience with a grain of salt: On my Besson with the small receiver, I use the Wick 1 small shank designed for it. My Bach 18, which has the standard American shank, only goes in @ 1/2 inch, effectively lengthening the horn @ 1 inch. But playing with my Bach 18 I have the same relative pitch as with the Wick 1, which has a really deep funnel cup, but roughly the same throat diameter. I do not adjust my tuning slide when going between the two.

A friend in Shrine band has a relatively newer M-W bell model, and uses an old Conn 2, and always complains about how far he has to pull his slide out. But since the German companies also make a lot of their horns at A=443 unless you specifically ask for the A=440 tuning, I have always chalked it off to his horn. But now I wonder.

FWIW, the 69C4 is one of the "brightest" and most shallow of the popular models out there. A mouthpiece with a deeper cup and a corresponding larger backbore will definitely lower the pitch a little, and you can push the slide in more. Of course the tone will be correspondingly different as well - darker and broader as opposed to brighter and more centered with the C4. The Conn Helleberg seems to be a popular starting point for many board members.

Get in touch with Doug Elliot. A few years ago he made an extra-long shank for me to use with my regular mouthpiece on a tuba that played a little sharp. He can match whatever cup and rim you like, and can make different-sized shanks that screw on. But be patient. Like any fine craftsman, he cannot be rushed.

I agree with trying the small bore tuba/funnel mouthpiece combination. A couple of years ago in community band, a guy played a small bore Conn and couldn't get what he wanted out of it with a Bach 18. He got a Conn Helleberg and everything improved dramatically: tone, intonation, tuning, etc. He gave me the Bach 18!

Speaking of Bach, this is from their mouthpiece catalog regarding mouthpiece throat diameter:

"NOTE: Numbers and letters are drill bit sizes. The smaller the number, the larger the throat. Generally, the larger the throat, the freer blowing the mouthpiece and the more volume possible. However, as the throat size increases, the upper register tends to sharpen and the player tends to tire more quickly."

So, in addition to looking for a mouthpiece with a deeper, more funnel shaped cup, you might consider looking for one that has a smaller throat as well.