Sharing app content to stage

marekcis - Could you please verify if you have required pre-requisites and given mentioned permissions in your application.
Meeting apps API references - Teams | Microsoft Docs

To use the shareAppContentToStage API, you have to obtain the RSC permissions. In the app manifest, configure the authorization property, and the name and type in the resourceSpecific field. For example:

"authorization": {
    "permission": { 
    "resourceSpecific": [
      { 
      "name": "MeetingStage.Write.Chat",
      "type": "Delegated"
      }
    ]
   }
}