I did briefly wonder about installing it in a container on the machine that's running my router, but a) it's probably not wise to put that extra burden on it, and b) I'm not sure Proxmox will allow it to access the HDMI output.
Having Done My Research, it would appear that a VM running in Proxmox can indeed have access to the HDMI port, and to Bluetooth for the controller. And given that the router side of things needs very little power, I think I'm going to try this.
Just need to work out how to go about it. And if it doesn't work, all I need to do is delete the VM and I'll have lost nothing.