Is it permissible to begin a supplication with "All greetings, prayers, and good things are for Allah" outside of ritual prayer?

There is no impediment to using the wording of the Tashahhud to praise Allah Almighty in supplication outside of prayer. Indeed, commencing supplication with praise and glorification is legislated and does not have a specific wording. If a person feels the greatness of Allah and their heart is present when reciting the words of the Tashahhud, then there is no objection to using them before their supplication.