This is not an issue to fix, it is a security measure to protect people's accounts. Wait for a week, then change your e-mail address and then you will be able to play.
You can read the explanation why in the thread I linked. If an attacker managed to log in by finding out someone's web password, they were able to just change the e-mail address to their own and the original user was not able to recover their account anymore.
I understand your frustration, but in one week, you will be able to reset your e-mail, send an unlock code to your new address and then play S3 forever.
At least you will change to a functioning e-mail address for the future.
What if you took a user agent string that is normally sent by a web browser and used it in LFS? I wonder if that would be enough to fool the ISPs to think it's coming from a browser.
While it doesn't work for you in game, I wonder if opening this link in Firefox works for you, as Scawen suggested few posts above. You can try changing the skinId parameter with different mod IDs and see if all work for you in the browser.
I wonder if it's Cloudflare blocking or slowing down traffic from some locations when it sees it's not coming from a regular browser due to the special user agent string. Maybe Cloudflare falsely thinks it's coming from a bot or a suspicious source. There may be a Cloudflare setting to allow all traffic through if user agent is "LFS".
Usually the mod author mentions it in the mod description in the mod details page. It's a common thing to use the extra light for it, but there may be exceptions.