The thing is, I can do the redirection manually. I can find the url redirection link in the retrieved web page, and use it to retrieve the web page I want. This works, but it is very long and ugly, and I would rather use the LWP::UserAgent redirection if I can.
Also, the page that I get back with the redirection loop is a page that can be redirected (it has the redirection url link), so I don't think that something is wrong with the website.