I've played sop in the past which isn't vastly different from D/Eb trumpet, only stylistically. I wouldn't profess to be the world's best sop, but could sit and play top-C's, loud and quiet for hours, if needs must. I've messed with mouthpieces and have come to the following conclusions which you may find useful.
Yes, a different mouthpiece can make an enormous difference in sound. The cup diameter, measured inside the rim, should be matched to your embouchure purely for comfort reasons. If you're like me, your embouchure's a bit 'fleshy' and so require a larger cup to 'get it all in'. Generally the rim profile is also about comfort and personal preference. You use the 'feel' of the rim to position the mouthpiece.
Now the important parts of the mouthpiece are cup, throat and backbore. The way these three varibles 'play' together is a bit of black art.
The depth of the cup dramatically affects the quality of the sound, deeper equals more fat, round and warm sound. But the depth of the cup can also affect the tuning of the instrument, not as a whole, but just certain notes, i.e. D, E, & G's. You may have to trade 'sound' for intonation.
The throat and backbone affect the perceived 'blowing' of the instrument. The 'back pressure' or the impedance to 'blowing' increases with volume and pitch. Increasing the width profile of the throat/backbore can reduce this impedance slightly making it easier to play louder/higher, and for longer. However, the impedance assists when trying to play soft and high, making it easier to 'key' notes. It's to play soft without playing 'against' something.
The attributes of the mouthpiece are all double-edged affairs. What you choose to play on is entirely down to the ability/preference of the player. You could choose the 'large' path but some extra effort would be required for the soft stuff, for example. You can choose a wider muothpiece making it more comfortable to play without sacrificing any sound. Let's face it, if god made man in his own image, god's not a sop player!