@tonywaghorn, I was able to run some more tests this weekend. I unmounted the projector so I could hook it up with a short, passive HDMI cable to the ATEM mini. Same behavior persisted. I had also purchased another projector (NEC PL-525UL) to see if it was projector related. When I hooked that one up with the short HDMI cable, the projector did not even register that an input was connected!
Next, I tried hooking up a HDMI to VGA converter and plugging into the VGA port of the projector. That worked - not HD, obviously, but shows the signal is getting there. That method worked with my longer 100' fiber-optic HDMI cable as well. It's the method we're currently using in the interim.
I did find this thread (
https://forum.blackmagicdesign.com/viewtopic.php?f=4&t=107747&start=50), which seemed to imply that a Decimator MD-LX unit would fix the issue. Was hoping not to buy another box, but I've got one on backorder with shipment expected in several days. I'll see if that works once it gets here.