Blue Iris

This would depend on the network a bit to say this is full proof but from what I'm gathering I think this would work....

You could try changing the subnet mask on the BI server itself to 255.255.254.0. While I generally wouldn't recommend having an incorrect mask, if both devices are on the same physical network (all wired up to switches) then that should allow the BI server to send ARPs for both the 192.168.0.x and 192.168.1.x while still maintaining internet access.

Because I'm a perfectionist I would probably only do this when you need to include a camera that is incorrect - so that you can get to it, move the camera to the right subnet - and then change the BI mask back.

*** FULL DISCLAIMER *** I don't 100% know your network so there 'could' be issues with this but I doubt it based on what you've said.

My network is a very basic LAN with a BI machine, a VM in Bridge Mode (its own IP address) for a Reverse Proxy, a raspberry pi, and a handful of cameras.

I was hoping there was a way to get the ARP requests for both to coexist but really wasn't sure.

I suppose 192.168.0 and 192.168.1 are only one bit apart, is that were you got your edit for the netmask from?

That should be a pretty safe change to make remotely and not risk knocking the whole network down.

Yes sir, thats where I came up with it. Right now your BI server thinks the camera is on a different subnet (technically it is) so it sends the traffic to the default gateway which then continues to route to the ISP (since it's also not a local subnet - though the traffic will die at the ISP). In your unique situation you can fool the BI server to have a larger subnet - with the tweaked mask - so that the camera is on the 'same' subnet. Again I would only do this hack to get the camera configured to the correct place and then set it back again.

Based on what you've said, this should be a totally safe TEMPORARY change.