Mail.app AppleScript to process POP3 mails stopped working in Lion
Judging from your description of the issue, it seems that Lion’s Mail.app has difficulties correctly executing a move AppleScript action on a PO3 server (or maybe just your POP3 server – not having a POP server to test myself, I cannot check). A solution could be to separate the two components of the move operation by replacing the line

move eachMessage to mailbox "Archived/BTR Monitoring Emails"


with

copy eachMessage to mailbox "Archived/BTR Monitoring Emails"
delete eachMessage


in the hope that an express deletion command will fare better. If that doesn’t alleviate the issue (because the deletions suffer from the same issue – I can’t say for sure from your question), you could also try setting the delete status of the message instead of sending it a delete command, i.e.

copy eachMessage to mailbox "Archived/BTR Monitoring Emails"
set deleted status of eachMessage to true


though I’m not entirely sure how that will go down within a POP server (it sounds like something designated for IMAP).

Finally, you could try commenting out the set read status of eachMessage to true lines in the filing and deletion blocks – this should, by rights, not affect the server, as POP3 has no notion of a “read status” (you need IMAP to store that server side), but it might conceivably be an issue if Mail’s AppleScript thinks otherwise. Setting the message status to read can also be achieved within a rule without resorting to AppleScript, so removing this from the script should be a non-issue.