I believe you need to convert the cookies to the format that BAS understands, and then import them into BAS using a standard method.
Take a look:
https://chromedevtools.github.io/devtools-protocol/tot/Network/#type-Cookie
right now i use GET on a website that works 24/7 so if i get an error then it means that proxy is offline.
but is there any more lightweight way to do this?
Not that I know of. You'll need to tunnel through the proxy to really check the connection. I either do a proxy checker site to make sure nothing is leaking, or I just do the simple GET request against the site(s) I'm working with/on.