Hello,
We don't require Kube-Prometheus-Stack for using the Azure Prometheus remote write sidecar. The document lists that as a pre-req because it was the most common method for our customers using Prometheus and we constructed the instructions based on that.
The Azure Prometheus remote write sidecar is basically a reverse proxy that handles Azure authentication to push data to Azure Monitor workspace. In order to use this, all you would need is to inject this sidecar to the same pod your Prometheus is running in and then configure your Prometheus to remote write to http://localhost:8081/api/v1/write
Furthermore, the Azure Prometheus remote write sidecar requires few additional values, which can be provided as environment variable in similar fashion as shown in the yaml in the following document