Hi Michael,
From your description it seems you are refreshing the logstream too often and you are being throttled as a result. When you use the portal and view the console logstream, the Azure portal makes requests to the API. The various Azure APIs have request limits, some of which are published while others are not.
Is there another way you can achieve your goal instead of relying on the console log in the portal? For example, can you send key status messages to another endpoint which logs to files, and then view the files?
Or, another potential is to view the log stream not via the portal but using a different method. For example, you would write small script that makes the requests to view the log in a more controlled manner (and maybe using different parameters) so as to not trigger throttling.
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP