Second monitor on Studio through KVM is always black

TL;DR: My new Mac Studio M4 won't send a signal to its second monitor.

I have a new Mac Studio M4 Max running Sequoia 15.4.1. I also have a Windows 11 laptop and two Linux desktop boxes. All 4 of them feed through a 4-port, 2-monitor, HDMI KVM switch so they can share my 2 desktop monitors (a 27" QHD BenQ and a 24" 1200p Dell). The Studio feeds its HDMI port into the BenQ monitor and uses a USB-C-to-HDMI cable to feed the Dell monitor (both passing through the KVM). The Windows laptop uses the same cable configuration as the Studio.

The Windows & Linux computers work just fine with both monitors. The Studio works fine on the BenQ, but it won't send a signal to the Dell. When I look in Display Settings, both monitors show up just fine. When I set the Dell to extend the BenQ, I'm allowed to move my mouse onto the second display, and I can move windows over there, but they display is always just black. If I set the Dell to mirror the BenQ, it's still black, but at least I no longer lose windows into a black hole. When I first power on the Dell, it shows a signal for a few seconds before it goes black.

I've tried a different KVM--same problem. I've tried a different USB-C-to-HDMI cable between the Studio & KVM--same problem. I've tried swapping the monitor cables so that the Dell is fed from the Studio's HDMI port and the BenQ is fed from the USB-C port--same problem--the Dell is still black.

I've tried bypassing the KVM and running the Studio's USB port directly into the Dell's HDMI port. That works fine as a proof of concept, but obviously isn't an acceptable solution because the other 3 computers need to use that monitor, too.

I tried replacing the Dell with a borrowed 1080p monitor, and that works fine. The Dell actually has no native HDMI port (just DisplayPort and DVI-D), so it instead accepts the HDMI cable through a DVI-D adapter.

What is it about the Studio that allows it to recognize the Dell monitor, but refuses to feed a video signal to it without a direct connection? Is there any kind of setting that I can change that will make this work? I don't have the cash to upgrade to a new monitor right now, especially since this setup works with every other computer I own except the Mac.