Honestly most player wouldn't be bothered with disconnecting and reconnecting their controller. I have mine plugged into the back of my computer and it's staying there. This means if I have to rely on that gimmick I won't be playing your game.
A better option would be pressing a button to disable/enable controller input, and then based on what state it is, treat it like you already do for disconnects/reconnects. This can be a button you press on the keyboard/controller to switch between the states.
This way the controller stays plugged in, you get your intended mechanic, and you don't have to have an issue with how the browser handles controllers.
You can still provide the option for them to disconnect/reconnect if they wanted to use that, but it being a requirement is detrimental to player interaction