looking in the ADMapping.daml I see this:
<customControl id="esri_mapping_downloadMap" caption="Download Map" className="ArcGIS.Desktop.Internal.Mapping.Ribbon.DownloadMap" disableIfBusy="true" showHelp="true" staysOpenOnClick="true" isDropDown="true"
...>
<content className="ArcGIS.Desktop.Internal.Mapping.Ribbon.DownloadMapControl" />
...
</customControl>
Note the staysOpenOnClick and IsDropDown attributes. I think those are what you are after.
You can find the ADMapping.daml here: <your Pro install location>/bin/Extensions/Mapping
and here: https://github.com/Esri/arcgis-pro-sdk/wiki/DAML-ID-Reference-ADMapping.daml