Nope, unfortunately not. Sadly, I do not have an active STA campaign.
I thought about the “auto-court-martial”, but I would recommend that this would only occur at 10 unspent shame. The system as I was proposing to modify works as follows:
- Player piles up up to 5 Shame that raises the complication range. It shall always be up to the player what to do with these 5 points of Shame. The player is already “punished” for piling up shame (as the complication range for rep checks is raised). Player agency is important.
- For any surplus Shame, the player risks intervention by others, i.e. by the GM or the CO or whomever. They, however, by spending Shame for them have to leave the player at least 5 shame.
If, e.g. a player has piled up 6 points of Shame, there would be one point that’s open to spend by whoever is allowed to spend it (you could also make it a decision of the whole group). Thus, a court-martial would be possible at 10 ‘banked’ Shame (5 that are to be spent by the player and 5 that are to be spent by the GM/whoever). There shall be an incentive to not to go over five banked Shame, but the player should always be respected in their choice to raise the complication range to 5. 
Well, it depends on how you handle the chain of command, but a promotion grants another rank and, thus, the character might “lose” superior officers (that are now mere colleagues) and might “gain” officers under their authority. In game terms, this is relevant for Directives.
Dunno. But since in the rules as written in the Klingon Core Book lack the option of saving milestones, you might choose to introduce something akin to the Reputation system. Maybe the players can start with 1 bonus Momentum into the next adventure? Maybe two or three Glory for one bonus point of Determination? I’d say I take one point of Shame away for two bonus Threat at the beginning of the next adventure…