Dispensation can be understood as "a distinctive arrangement or period in history that forms the framework through which God relates to mankind". Under both the Judges and the monarchy period (including David's dynasty), the judges & kings are "state" authorities charged with secular implementation of the framework necessary for the priestly system and the constitutional (the Torah) system to operate. David would then fall under the yellow "Human Government" box in your diagram.
I would compare Israel / Judah king's role at that time to how a Christian Roman emperor (like Constantine or Justinian) or the King of England see their role in giving the Church with proper secular support and secular powers to exert her spiritual authority, just like God expected the Israel / Judah kings to build and support the temple, to support the right judicial authorities, and to cry out only to Yahweh for help during times of invasions / famine, etc. But we read in the books of Samuel, Kings, and Chronicles how with the exception of David the kings compromised their duties from little (like Josiah, Hezekiah) to huge (like Ahab). They are supposed to destroy foreign god's temple and to implement justice (Torah) but INSTEAD they import foreign gods or by not going far enough to destroy foreign god worship erected by their predecessors. So God responded in the same degrees: respectful burial for Josiah / Hezekiah, relatively minor troubles for them during their lifetime, but total humiliation and the end of his dynasty for Ahab, culminating in the 2 exiles under Assyria and Babylon.
After returning from exile under Persia, throughout the second temple period, some (most?) Jews expected two kinds of Messiah: a priestly one coming from the line of Moses (Levite), and a Davidic one. Jesus was both, even though Jesus descended from David. I think that is why the book of Hebrews justified Jesus's office as Priest by saying that his priesthood was from the order of Melchizedek (because He was not Levite by blood). It is Jesus's Priesthood which is part of dispensation, hence the purple Grace box in the diagram. But Jesus is also King, the King of Christendom until today, continuing until the second coming, but this authority falls under "Human Government" (yellow box).
So yes, David was a big deal, and so is Jesus's authority as King today! All countries's supreme authorities dominating the news recently, whether it's US President, Canada Prime Minister, King of England, Japanese emperor, Saudi King, Thai King, Chinese communist party, President of Russia, should all submit to Jesus's authority as King in the yellow box to give room and support for the Church, for example by defending her existential rights, her secular rights, her input to the country's laws, and police protection against violence.
But dispensation is essentially about our relationship with God, administered in the current dispensation through Grace (applied directly to individuals for Protestants, or through the sacraments of the church for Catholics), and I think that's why David didn't get a separate marker in the diagram but he definitely belonged to the yellow box and all authorities of the world today should look at David for role model.
Now to answer your specific points:
The monarchy represents a major shift from a judge/prophet-led and distributed nation to a centralised king-led nation, and from the priests being the people's sole representatives before God to the king also being their represenative.
I would not call David as God's representative, but as God's choice (as in God chose David over Saul). In the monarchical period I believe the King was supposed to be anointed and coronated by a priest, God's true representative. When a king needs to seek God's help the king needed to go to the Temple, a precinct of the priesthood. So a King didn't have inherent access to God, unlike a priest. God also used prophets as official messenger, the "mouthpiece" of God, and the King was supposed to listen and obey. But prophets do not have stable, permanent office like a priest, and anyone (non-Levite) can be called to serve as a prophet. The Pharisees and Sadducees were fine with Jesus if he restricted his activity as a prophet, but Jesus made bigger claims.
David's covenant is one of the few Biblical covenants, the others (Noah, Abraham, Moses and Phinehas, the New) are associated with the beginning of dispensations.
Remember that David's covenant is not the same as Mosaic / Sinai covenant, and is not the same as Abrahamic covenant.
David's covenant is personal (to David alone, the man after God's heart), and God fulfilled it in Jesus who now rules forever. It's a done deal; there is no more to be expected.
But people are not saved by David's covenant, people are saved through the Mosaic covenant (the cyan "Law" box), also a done deal since Jesus fulfilled the Law.
Abrahamic covenant is also different in kind, having to do with "physical stuff": descendants, prosperity, and a means to lead Gentiles to God (some would say that's why God is preserving the Jewish people as a race).
So it's best not to confuse covenant with dispensation.