It sounds like the issue is that HDMI doesn't carry power like USB-C can. What you're looking for is an adapter with a USB-C power passthrough. That way, you can feed power to the monitor while also getting the HDMI signal. The adapter you're looking at seems like it might work since it has a PD (Power Delivery) port.
Just make sure it supports enough power for your ZenScreen and that the PD charger you're using is strong enough.