I think you could feed a cookie in (and was how I was thinking of doing it), but the problems are:
- the cookies are tied to an IP and change in a serial manner
- every now and then LJ just logs you out
- if you have a slow connection you can see in the status bar that it goes through a bunch of redirects

None of those is insurmountable but it makes for a fragile system that will need constant maintenance of that cookie.