Hi Rob,
I'm not familiar with Moodle on Windows Azure, but running the (automated) backup is a processor intensive task indeed.
We are running our Moodle instance on Redhat Linux + MariaDB database+data on a separate server with 8 CPU's and now
96GB memory for the database. We also had problems to tune this process and with a lot of courses, it takes a lot of time.
What is the number of courses backup-ed?
Things to look at are the temporary backup folder (fast enough FS?), load balancing and caching (do you use a caching server
on your Ubuntu OS?), just to name a few.
Ken Task wrote some good articles about the backup proces here in the forums, just have a look at them I can only say.
Maybe this thread can give you some insight, I had problems myself with the automated backup:
https://moodle.org/mod/forum/discuss.php?d=383012
by Alain Raap.