I found a seamless switch at a reasonable price and that stopped the video from being dropped. It's a Goronya HDMI 4X1 Quad Multi-Viewer Splitter.
Thanks for your advice.
Sure, I'll keep you posted. So far we've used the Goronya switcher for two meetings with no issues. The switcher can do up to 4 HDMI inputs. We are using only two, the camera feed and computer feed. It's nice to know that if the person running things accidentally presses the button for either the 3rd or 4th feed, the switcher outputs a black screen and there is no drop in signal.
The side-by-side 50/50 image work fine, but I doubt we'll use it much. You have a ton of picture-in-picture options on the remote, but to get to the one you want, you have to cycle through all of them. That would be too distracting for those watching from home.
I don't recall seeing any black bars and some of the split options resulted in distorted/stretched views. Again, we decided not to use that feature because there is no quick way to get to the split you want without cycling through all of them. For the person watching at home, this would be frustrating I'm sure having to watch us cycle through.