How can I stop an HDMI switch from sending an unsupported signal?
After suggestions in the comments, I tried an active adapter instead of a passive one.  While the adapter may have been flakey, for some reason, Amazon deliveries for the past week have been completely inept and I wasn't about to risk dealing with that again to return the adapter and get a replacement.  Instead I went to Best Buy and got a different HDMI switch.  I'm not going to link to it because, although it solved this problem, it had another issue.

I'll be trying another adapter when it arrives and if that works, I'll post information on it.

In the meanwhile, I was using a Kinivo 3 Output HDMI switch.  While it handled the normal DVR and BD player video signals, it was not able to handle the video signal from either display adapter from my iMac.  Be aware of that issue if you are looking at that brand of an HDMI switch.



EDIT: I have to add that, after experimenting, while I was using a switch that worked for my display adapter, I found an issue with the connector.  Bottom line: The connector had a very narrow "fail range."  Most of the time it was plugged in, it'd work, but there was just one angle that would make it fail.  Of course that angle was the angle it was in when in the Kinivo switch, where the signal would not go through.  Once I swapped it with another cable, it worked fine.