When I am configuring Redgate's Sql Azure Backup tool to restore a database, it doesn't let me choose anything from my local Sql Express instance. Any idea why?
I've not done any testing on sqlexpress databases which may well be your issue. I've had to deprecate the product in favour of a hosted solution at http://cloudservices.red-gate.com. Perhaps you can try SQL Compare and SQL Data compare as SQL Azure Backup was based on the engines of those tools.
I wrote SQL Azure Backup so any questions just drop me a line.