I am not sure about the REST connector approach: in any case, to get oauth2 to work, you'd need credentials. If your client's Azure administrator doesn't grant them EA access, would they set up a similar thing for REST?
If the report you're trying to download is associated to a Subscription, you might not get all the relevant data. Somebody who knows more about Azure, please correct me if I'm wrong.
Furthermore, there's a lot of logic inside CT to map the Azure billing data. Some other things might not work, although you might work around this particular issue.
On a more positive note:
In trying to use the REST connector—albeit for something slightly different—we tried with Postman, to ensure a better troubleshooting while things weren't working. Apptio Support also supported this. If you can get to your data via Postman, you should be able to get the REST connector to work.
#Datalink