This is confusing because you don't mean rim width when you say "rim width". You mean mouthpiece diameter. The rim width would be more on the order of 8mm (the width of the rim; not the width of the mouthpiece).
That being said, the appropriate mouthpiece diameter (and rim width, and backbore, and throat, and ...) depend very much on the player. I prefer a quite narrow diameter, exactly along the lines of 30.00-32.00 mm. So I use a Wick 5 on my Eb and prefer something like a TU 17 or Schilke 66 on my BBb. The Wick 5 has a 30 mm diameter and it works very well for me on the Buescher Eb.
I don't at all understand the business about bleeding since I have never had any bleeding, no matter what mouthpiece I was using. I can't imagine what would cause bleeding other than some particular medical condition (e.g., taking a blood thinner like Coumadin) or using excessive pressure against the mouthpiece.
I did try a Wick 3 on my Buescher Eb and found it to be MUCH larger (in terms of bowl volume) than I expected. I may try one again in the future, and perhaps a 4 as well. But the 5 works extremely well.
Gary Merrill
Wessex EEb Bass tuba (DW 3XL or 2XL)
Mack Brass Compensating Euph (DE N106, Euph J, J9 euph)
Amati Oval Euph (DE 104, Euph J, J6 euph)
1924 Buescher 3-valve Eb tuba (with std US receiver), Kelly 25
Schiller American Heritage 7B clone bass trombone (DE LB K/K10/112/14 Lexan, Brass Ark MV50R)
1947 Olds "Standard" trombone (Olds #3)