I was using session based. From what I gathered by reading on forums, the Flash player sends IE cookies, no matter what browser you were using. Therefore, if you use IE it's fine, or if you're IE browser maintains the session after closing it may also work.
SWFUpload had a PHP workaround for this session bug, but I couldn't find an easy way to do this in Domino, so ended up re-queueing the first file so that it:
1. Sent the file to a public access page along with the LTPAToken to set a Domino session for the Flash player
2. Changed the upload URL
3. Sent all remaining files plus the re-queued first file to the new, correct URL.
This worked, but is a pain, as if you are uploading a large file first it has to upload it twice to store it once.