Anything that begins with "I have heard that ..." cannot be regarded as a good reason for anything.
One reason for this is that you can as easily hear exactly the same thing about at least 200 other mouthpieces -- just from different sources. So let's kick that "reason" to the curb.
Trying solve what you see as a fatigue problem by chasing it with hardware alteration is generally a very bad idea: the new mouthpiece just turns out to be a crutch to get you a rapid gratification of being able to play longer. This does not result in developing the long-term and persistent skills and habits that you want as a maturing musician.
Honestly, it's never occurred to me to target buying a new mouthpiece so that I could play longer on it. I'm more concerned with the quality of the sound, flexibility, and range it will get for me over time -- and consistent with proper and good skill development. Stamina will come through that sort of development. Note that this isn't to say that some mouthpieces won't be better or worse for you than others because they'll permit better stamina in the context of your development. But to state as one of your two primary goals an increase in stamina is probably indicative of a misperception of how you should proceed with your development -- kind of like expecting the cart to pull the horse. It's like the people who -- instead of developing their embouchure to play in a higher range just get a smaller mouthpiece. Presto! They're there! Um, yeah, sort of.
I think that if I were you I would spend some effort with a good teacher and discuss how you want to improve over a particular period of time (whether it's weeks, months, or years) -- and then get some good recommendations of what a good mouthpiece choice would be for you based on his own knowledge of mouthpieces and his experience with your current playing and goals. Otherwise, the danger is that you can pick up some bad habits that could end up taking a long time to correct.
If you can't get good lessons (from either a euphonium player or a low brass player familiar with euphoniums) locally, there are people (e.g., Doug Elliott) who give lessons by Skype. And you wouldn't need to be committed to a lengthy sequence of lessons in order to learn the best direction for you at this time. More like hiring a really good consultant to help you address a couple of problems instead of embarking on years of lessons. Just a thought.