I figured it out, and it surprisingly is possible.

ChatGPT was able to walk me through it. The big issue I had was that I was trying to connect using my own machine, which does not work. However, trying to connect using my phone did work. I also had to bind my VPN's local IP (it's different than the one that displays in the app) for nginx to work.

I could verify my VPN's local IP with ip -4 addr show. It's the entry with tun0.

http {
	server {
		listen LOCAL_VPN_IP:VPN_PORT;
		server_name  localhost;
...