Two parts to the solution:
One (as above) was a flaky HDMI-USB converter that was causing dropouts - replaced that, signals are now stable.
Second was an active HDMI splitter that can send different resolutions to each receiving device; I can now feed 1366x768 to the projector, and 1920x1080 into OBS via the (new) HDMI-USB converter.
One (as above) was a flaky HDMI-USB converter that was causing dropouts - replaced that, signals are now stable.
Second was an active HDMI splitter that can send different resolutions to each receiving device; I can now feed 1366x768 to the projector, and 1920x1080 into OBS via the (new) HDMI-USB converter.