We tried JMeter before and thought that it was okay, but it's
Just a tool, it is up to you to get it running somewhere: not enough for us (on its own)
So Azure Load Test looks like it might be that "somewhere".
I followed the simple tutorial to upload the test plan from the original investigation. I adjusted it to use the start URL from a recent load test run, and set 40 users, infinite loops, and duration of 600 seconds, to be comparable to our regular load test.
It costs €8.998 per month (includes 50 Virtual User Hours per month) + €0.135 for each additional Virtual User Hour.
The results look quite promising: