ZeroTier through another VLAN

Hi, I’m new to ZeroTier, I configured it well on my most devices. But I have one particular and don’t know how to do it. Here is how thing goes.
A typical internet process looks like this.

image (I can upload only one media… please check the online word document)

The company I work for uses university’s ground to save rent. The broadband internet here is limited to teachers only. Other people need to go through an expensive and complicated authorization process. We currently only have one Linux server on the whitelist, and we must bother teachers every day to get working computer to use broadband internet. I figured out a way to avoid authorization. Full process looks like this.

I use tap windows adapter (virtual network adapter) to communicate with the whitelisted machine ( to get internet access.
Now I want to expose the to ZeroTier VLAN, but I can not get through. But it connected for just one second and cannot connect any more (REQUESTING_CONFIGURATION).

I think if it works, full process may be like this.

There may be other solutions, such as adding a proxied gateway, and connect to the gateway, and use ZeroTier.
I hope I made it clear, if not, contact me and I will reply as soon as possible, any suggestions are appreciated.