In your setup, Docker is mapping port 8000 on your Docker instance onto port 8000 on your machine, which is what enables you to access the Splunk UI of the Docker instance. The -p "8000:8000" part of your initial Docker run command is what did this. You need to add another port mapping flag for 8089 to allow access to the REST API, eg -p "8089:8089" to allow access to the REST API via your local machine. So the whole Docker run command would be something like:
docker run -d -e "SPLUNK_START_ARGS=--accept-license" -e "SPLUNK_USER=root" -p "8000:8000" -p "8089:8089" splunk/splunk
Now you should be able to access the REST API via https://localhost:8089 (nb, you need to use https, not http for REST API, unlike you did in your original post)
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