As far as I’m aware that’s currently not possible with tiled. However you can write a batch file, and then call that with the custom command. In the batch file you can first let tiled export the map (see https://doc.mapeditor.org/en/stable/manual/export/#export-formats for the command line arguments) and after that you can call your game.
Even in the development version with scripting support, I don’t think this use-case is supported right now. While you can trigger the export from a script, you can’t currently trigger custom commands. That’s definitely something I’ll want to cover before the Tiled 1.3 release, so thanks for bringing it up!
In the meantime, @EJlol’s suggestion should do the trick!