Hi,
You are asking a really good question. Unfortunately, I don't have an answer for it and would recommend to ask Support via the https://support.smartbear.com/message/?prod=LoadComplete form. I will appreciate it if you update this thread with their reply and/or your findings.
LoadComplete supports Amazon cloud (https://support.smartbear.com/loadcomplete/docs/testing-with/configuring/cloud/index.html, https://support.smartbear.com/loadcomplete/docs/use-cases/load-testing-on-cloud-computers.html) but the documentation says nothing about Azure. I would expect that the same approach that you used for in-premises testing should work for the cloud but only if you have permissions for the machines where the target web server is installed. If you don't have appropriate permissions, you will not be able to connect to the box with the web server and will not be able to get the values of performance counters. You still should be able to get performance results of your load test from the end-user point of view (e.g. page load time, data through-output, etc.) but will not be able to correlate these results with the server-side data and it will not be possible to determine whether performance degradation is because of exhausted server's memory, or CPU, or disk overload.
This is how I see the things...
