Perhaps it is just that we have been exploring these two topics simultanously, but I think there is an interesting relationship between them.
I wrote about bitachon that
Bitachon is awareness that the A-lmighty is acting in a covenental partnership with you. It is from there that Rav Dessler’s formula for hishtadlus emerges, one partner only picks up what he does not expect from the Other’s contribution. It is the Chazon Ish’s awareness that every event in our lives is part of a plan. And yet we can avoid simplistic dismissals of suffering. Yes, Nachum Ish Gamzutells us that everything is for the best, eventually. But since I must remember that no story, no “eventually”, is ever over, I can not find meaning or redemption in that fact. Pain remains pain. And yet, having bitachon demands that trust in “כחי ועצם ידי — my strength and the might of my hand” is misplaced, and through my activities I can not avoid the tragic. It is part of the role I play in the Divine Plan, and to not accept them as from Him and part of the covenant would be disloyalty to it.
And the melukhah of Malkhios can be seen as
אֲדוֹן עוֹלָם אֲשֶׁר מָלַךְ בְּטֶרֶם כָּל יְצִיר נִבְרָא
לְעֵת נַעֲשָׂה בְחֶפְצוֹ כֹּל אֲזַי מֶלֶךְ שְׁמוֹ נִקְרָא
Eternal Master Who was King before all things were created
Once He, with His Will, made all, then his name was called “King”.
In Shema, we are referring to “asher Malakh”. On Rosh haShanah the goal is to make that manifest in this world – “azai Melekh shemo niqra“. Not the theory of Kingship, but actually declaring Him as King. “Hashem E-lokeikhem” even before we reach the point of “Hashem Echad“.
Hashem is unchanging, He was King in some ideal sense even without creation. But to be a king, “ein melekh belo am – there is no king without a nation” declaring Him their King.
A Melekh need not impose His will in the same way that a Mosheil does. A Melekh, therefore, has the opportunity to act with kindness and mercy at times when a Mosheil could not. We therefore introduce High Holidays, the days of judgment, by declaring G‑d’s melukhah. By voluntarily accepting Him as king we obviate the need for G‑d to direct us on the right path through trials and tribulations. The point of Rosh haShanah is accepting Hashem as our Melekh not just in theory, but declaring our acceptance of His Reign, thereby changing His relationship to us from one of Mosheil to that of Melekh.
Comparing the two, it would seem that both are about the beris, the covenent by which Hashem is our constitutional Monarch. Bitachon is the trust we have in His contribution to the beris. In Malkhios we declare our willingness to contribute our share.