Several things to unpack here, but I suspect the way you're currently doing it (separate live stream with Zoom) is the best way to go given the circumstance. Not sure Office 365 will have much to offer you here.
Yes, there are ways to do things "once" and then send that stream to other places (e.g. Twitch streamers use OBS to do cool stuff like this), but I suspect that this is a bit beyond what you're willing to invest time-wise.
Churches are starting to use this, as well. This guy installs it and walks you through it, but there are some additional steps that you'd have to do to get it customized for your environment.
As for the Zoom WiFi problems, these can likely be fixed if you move to a laptop with a wired connection. The built in cameras are often not as good as smart phones, so you might consider an external one. Feel free to drop me an email or phone call if you have any other questions. firstname.lastname@example.org 408 217 1505