What triggers asking connected device to reconnect to WiFi router?

Solution 1:

First of all as you seem to be interested in this topic, I will give you a general explanation of what you are trying to set up, as this will make it easier for me to write my answer and you are learning something in the process.

What you are calling Router A in your question is in fact the only real router in your setup. The Router B will become (or is in fact already) a so called Access Point or short AP. It's sole purpose is to translate/convert the incoming network traffic or signal over the ethernet cable to a wireless signal that will be transmitted via Wifi. In comparison to an AP, a router is a management unit in your network. It assigns more or less unique addresses to all devices that are connected to your network, no matter if wired or wireless. Only by doing this, you can reach out to a single device without sending the same data to all devices and hoping that the wanted device is the only one that answers.

Now to the part where you need to configure some settings:

The only things other devices in your network will see are the SSID and BSSID (if connected wireless) or of course the hardware address of your used LAN port if connected via ethernet cable. Everything else inside your router that you as an end user can configure via a web interface is kind of a black box to all other devices. Meaning that most of your devices won't care what you do with your network infrastructure as long as they get an IP-address in order to communicate with one another and the Internet.

One last thing, in my network I have multiple devices that I cannot access that easily as well, and when I need to make changes to the network, I can connect to them via another web interface and just load another profile I can create there, in order to make them connect to the new SSID after a reboot. Then I can do whatever I want with the routers or APs and as long as after the procedure the new SSID matches the one I set up in the profile, they will connect to the new network. Maybe that works for you too, just take a look into the manual.

tldr:

As long as you do not change the SSID to something new (something your CCTVs don't know i.e. the old one of your Router A) there should not be a problem.