Error reporting in Azure Functions vs App Insights - azure

Question

We call REST based APIs hosted by Azure Functions and fail to implement a consistent error handling supporting App Insights and wonder what can be done about it:
If we don't handle exceptions of the function, then App Insights
reports a 'failure', but the service returns only the the error code to the caller, but no error content:
Hence, the client receives a 500 and thats it.
If we handle the exception and log it (to AppInsights) then App Insights stops reporting a 'failure' hence monitoring on function level is broken. We can query for the exception, but they are out-of-context (i.e. we can see the exception by a custom query only) and we don't know which function is impacted actually.
How to marry up the two needs:
Let the function fail so that AppInsights reports the failure (and monitor can alert)
Return a bit more meaningful error message to the caller than 500.
Example on how it looks in AppInsights:
Exception is visible on the Exceptions tab, but the underlying operation has not failed
UPDATE:
According to Microsoft, App Insight Failures are exclusive to unhandled exceptions. Still, open whether there is a way to at least pass-through an error message.

Answer 1

If don't handle exceptions of the function, then App Insights reports a 'failure', returns the error code, but no error content. Hence, the client receives a 500 and thats it.
App Insights doesn't return anything, so what do you mean with returns the error code?
If we handle the exception, LOG it (to AppInsights), return simple error message in code 500, then App Insights doesn't log this as 'failure' hence monitoring is not possible.
Can you show how you do the logging? Because as soon as you log an exception using App Insights you should see it under failures.
This should work:
try
{
...
}
catch(Exception e)
{
logger.LogError(e, e.Message);
return httpRequest.CreateResponse(HttpStatusCode.InternalServerError, e.Message);
}