Yes, source mode is the key. The Properties UI is not a normal editor area, so template insertion commands usually do nothing there. I would insert it in Source mode and keep the template output valid YAML, for example no unquoted colon characters inside a value. If you are using Templater, test with one simple field first, then expand it, because a small YAML syntax error can make the properties panel look like the insertion failed.
When you insert a template into the active note, all the properties from the template will be added to the note. Obsidian will also merge any properties that exist in your note with properties in the template. (Templates § Template properties - Obsidian Help)
As CawlinTeffid and others said, Source mode is the key — but the real trap is the template output being invalid YAML. An unquoted colon inside a value, a tab in indentation, or an unescaped [[wikilink]] in a property will silently break the Properties panel and make it look like the insert failed.