Welcome to the Bose Portable PA Community.
Thanks for all the details.
First suggestion: Set everything up before you get to the show floor. Figure out how everything works and learn to deal with any issues in advance.
Take extra batteries!
Put the S1 Pro on the stand so the talker with the microphone will be behind it (ideal) or beside it (okay).
This will be a challenge to the talker because s/he may not be able to hear himself very well, so he'll have to trust that it's all working. There's a little stress associated with that, because when a talker can't hear himself, he tends to talk louder to compensate. That's good for the microphone and gain before feedback. It's just harder on the talker's voice.
Aim the S1 Pro at the listeners. I know, that sounds obvious now, but you'd be surprised how often people don't do it.
Turn up the S1 Pro channels only for the microphone(s) you are actively using.
If you got the directional version try the Cardioid tip or Hypercardioid tip. It's hard to predict which will work best. It depends on where the talker will be in relation to the S1 Pro.
Set things up so the E1 microphone is not pointing at the S1 Pro. For example, if the talker is wearing the microphone over the left ear, the microphone is pointing right. Put the S1 Pro in front of, or beside, and to the left of the talker. Try all three ToneMatch Presets to see which sounds best.
If you use the hand held microphone - hold it as close to the mouth as possible. (Within and inch or two, not 10-12 inches). Closer is better. Talk into the microphone, not across it. Use the ToneMatch Preset for microphone. Be aware where you point the microphone - never at the S1 Pro.
The Shure Lav microphone is the most likely to be a problem. It's hard to get those close enough to the mouth to get good gain before feedback.
If you get feedback, turn down the volume, or turn down the treble a little.
Also - if it's obvious that people are having trouble hearing the talker, invite them to come in closer.
Does that help?