(See the best response by Nadrew.)
I have spent a grand amount of time trying to work this with zero to zero luck.

TGStation hasn't been functioning properly.

My current setup involves me hosting a TG server on a computer seperate than my own, Others able to successfully connect, But myself, a resident on this network with zero firewall restrictions clientside or serverside existing, have not been able to connect to the very server I am hosting without having to join through Dream Daemon there.

Is there a way to how I can use my BYOND client to connect to my server?

(This has probably been asked by alot of people but everyone is lonely nowadays when it comes to IP troubleshooting so ripper roo)
are you using your external IP address, or your internal IP address to join?
you can't connect to a localhost using an external ip in byond

it's kind of a bug tbh; byond should at least be able to identify if something is locally hosted and automatically switch the ip for you. I'm 99% certain you used to be able to do this and somewhere along the way it broke
Best response
That's actually the fault of your router, not BYOND. Some routers handle the loopback that happens when you connect to yourself oddly unless you use an internal IP, and others do the same unless you use an external IP.
So I just use the internal IP of my machine say its, What would the link dreamseeker would open with be?

replacing port with the port you are hosting out of.

If the computer thats hosting this is windows based, you should be able to replace the ip with the machine name of the computer.