Can't you just setup a 2nd database instance and run your conversion from that. You could do things I have in the past and create a 2nd instance of your DB. Copy all the DB files from the current production instance to that 2nd instance. This will allow you to call that 2nd instance (it has it's own user name and pw) completely autonomous from your production site.
You then create a 3rd instance to receive all of the converted database files of which you will make the primary site when completed and tested. You can suspend posting via security for a a while and still allow access browsing of the original production site while all this is going on.
Once things are tested then you can cut over to the new site easily.
Just a thought. Your hands aren't really that tied. I am sure you can get the host to help out with the initial copy of your production DB to the secondary instance. Worst case you can sqldump the production DB and import it into the 2nd instance DB as well. Embrace this change as I know you immediately benefit from it. I wouldn't fear the process.
I guarantee the process will be much more obtrusive using a non UBBThreads software. I can also guarantee you will lose information during that non UBB conversion process.
I would hate to see you move to lesser boards. I get nothing for saying this and I have purchased and have owned the other products. You're with number 1 in my mind right now.
