First of all, I apologize if this has been covered. I searched a few key words, but none of them seem to be related to this. Below is my church's setup. I know it isn't anything like perfect, but we, like all churches, are on a budget. The church was at one time five suites of an office building. For necessity, the stage and the chairs are a semicircle. There is a projector which serves the center and a little left/right of center of the stage. The far right and left are served by televisions. For the Worship the TVs carry the proclaim feed. During the message they show video so those people don't just get to look at the Pastor's backside. I am going to start with the flow of signal, and then I will finally get to my question.

Camera goes to a 2 Port Splitter. 1st port goes directly to the PC running VMIX for live stream. 2nd port goes to HDMI switch. One input is the camera feed and the other is the Proclaim feed.

When we switch the feeds to the side TVs from Proclaim to Live Video, I get 1-2 seconds of No Signal displayed on the VMIX input. I understand that CEC can be a problem, and I have disabled it on the TVs. The switch is not, by any means, awesome, but it wasn't the $10 version either. It is also active. Has anyone run into this? Was it the fault of the camera, the switch, or a setting that I am just missing somewhere?

I am trying to avoid having to spend the money on an professional switcher if I can help it. Also, if it is the camera that is at fault, it would be a waste of money anyway. I do know that we have used two varieties of cameras and it did it to both.

Any advice is more than I had a moment ago.

We do have a somewhat similar setup in our church, with a projector for the audience and a live-stream for web and external locations.
Is there a reason why you don´t feed your projector and TV´s from the vMix pc using an external output? This way you could completely get rid of the hdmi switch, because all switching would be done in vMix.
The latency from vMix is the only reason I could think of why you wouldn´t run this setup, is that the reason?
Other than that it does only provide benefits do do it like that, you can for example overlay verses from the bible over the video feed of the pastor or similar.

We run a setup like that and are quite happy with it.
For more convenient control of the external output we use a midi controller.