words - What is the function of "Eloheinu" in blessings?

Many blessings open with the familiar formula:

ברוך אתה יהוה אלהינו מלך העולם

In additions, many blessings close with the familiar shorter formula:

ברוך אתה יהוה

Now, we know (Berakhot 40b, Rambam Berakhot 1:5, ShA OC 214) that a proper blessing needs to contain mention of both God's name and His kingship (Shem uMalkhut). Clearly "יהוה" is a part of the former of those requirements and "מלך העולם" is a part of the latter. Where does "אלהינו" fit in?

On the one hand, אלהינו is a name of God which cannot be erased (Rambam Yesodei 7:2, ShA YD 276:9) so it seems to be part of the "Shem" part of the formulation. Mishna Berura (214 sk 4) rules that אלהינו alone can function as the Shem in a blessing. Indeed the combo "יהוה אלהינו" appears quite frequently in Tanakh (with conjunctive Trop on the first word), and, even when unwarranted, people automatically run them together.

On the other hand, in the shorter closing formulation of longer blessings the chosen name of God to use is not "יהוה אלהינו" but just "יהוה". I can accept that Malkhut need not be mentioned again at the conclusion, but why should only part of the Shem be left out? It's worth noting that the name אלהינו has associations with Divine Judgement which in a certain sense fits more with Malkhut than Shem.

I note that in many prayer books there are Trop marks on the blessing preceding the Haftarah, opening with בר֨וך את֤ה יהוה֙ אלהינו֙ מ֣לך העול֔ם which, by putting the primary division on יהוה, indicates אלהינו is part of the latter phrase. However, these notes are of unknown origin and clearly do not conform to the regular rules of Trop (you can't have two Pashtas in a row unless they follow a Revi'i; I suspect the notes were just lifted from Chronicles 1:29:10 without much thought to context). Hence, I don't think this should be taken as conclusive evidence.

I note as well that in my experience, those who say "Barukh Hu Uvarukh Shemo" when hearing God's name in a blessing (cf. OC 124:5) do so after hearing the word יהוה but before the word אלהינו (eg. during the Chazzan's recitation of the morning blessings), indicating a separation in the blessing. As before though, this could be a carryover from, for example, the repetition of the Shemoneh Esrei where that phrase is regularly used in the context of the shorter closing formulae of blessings.

So which side is it? Is "אלהינו" in a standard blessing part of the Shem or part of the Malkhut? If part of the Shem, then why leave it off in the closing formula and pause between the two halves of the Shem? Proofs from earlier sources (≥Rishonim) are especially sought.