As far as I know, you can’t login using cookie on another IP. But I notice that some applications, like Blau Quiz Center are able to use ranking when someone passed the quiz.
Is there any way I could use the username and password login instead of cookie?
To my knowledge right now, one way to bypass it is by logging in from a European IP address, getting the cookie and then hosting the ranking somewhere in Europe as well. (There have been rumors that it’s actually not invalidated by IP change, rather by continent change). This is what myself and my friends have done as a temporary fix and it seems to work fine for me, but obviously it won’t work forever if Roblox is going this route with rolling it out permanently on the basis of IP change.
If you’re asking for a specific way to login and you don’t want to host the server in your continent nor use a VPN to login where it’s being hosted, you could always try logging in via Roblox’s Authentication API and get a cookie from that. The issue with doing so is captchas though, meaning you’d have to use some sort of captcha solver implemented to handle captchas.