Clicking the button doesn’t proof that you are a human. All the checks happen way before you even click the button (or sometimes even before visiting the website). Google also offers a similar button for their users and since cloudflare is also used on almost any website, they have a lot of data about you. They check your cookies, browser agent, device, settings, your IP address, if you use a VPN or proxy, etc. If you visited other cloudflare websites in the past with the same device or IP, and so on. So they know you and your device way before you even click the button. This is also the reason why you sometimes see a robot arm (made of Lego) clicking the button, and is still recognized as human. But as soon as you use a different IP address or a VPN (or even use a shared IP address, like in your company’s network) you have to solve CAPTCHAs. Of course they also check mouse movement, but this is only one part of many checks.

Some proxies block Websocket Connections by default for unknown URLs, even for port 443. Don’t ask me how I know :D

You can setup Apache Guacamole on your server. It uses Websockets by default, but it also has an automatic fallback to plain http/https. It will be ultra slow, but at least it will be working. It will behave like any other website, so no security risks for your company if they already have a proxy server to monitor your Internet traffic.