But I don't get why a web archive of the login page can be a "solid evidence" since logging using the alias in the achieved page wont work.
It's serve as proof that recover Bitcoin by alias is an option in case the customer service giving OP hard time (either because ignorance or don't have serious intention to help).
IP? I doubt anyone keep history of IP given by their ISP
You may be able to find it somewhere: some websites send an email with your IP address when you login, and many forums keep IP addresses with posts. You may be able to ask on some of the websites you've used.
It's interesting idea, but a bit difficult since ISP usually give dynamic IP which usually changed in few days.