Word order: "dear my lord" in Shakespeare

In general, the form appears to always have been "my dear X". It is now (unless you're writing a letter), and it was before Shakespeare's time. This can be seen in examples from Middle English, from this entry in the Middle English Dictionary (emphasis added):

And dele A-mong my Frendes · and my deore children.
_{The vision of William concerning Piers the Plowman (c1390)}

Luke it be done and delte to my dere pople
_{Morte Arthur (c1440)}

My dere sonne, I ȝef vp my sowle ynto þi hondes
_{Mirk's Festial (a1500)}

There are also examples of "my dear X" in Shakespeare's works, as can be seen here. So why "dear my X"? Hot Licks suggests that we "[c]onsider that "m'lord" is a title". I did find some evidence that "dear my lord/lady" was used by others, e.g. here and here (but neither of these are particularly close to Shakespeare's time).

However, the bigger problem with this theory is that Shakespeare uses "dear my X" when "my X" is not a title (e.g. "dear my brother"), as you can see here. Looking into this, I found a possible explanation:

Shakespeare occasionally uses a peculiar idiomatic phraseology similar to that employed in the Italian language. He sometimes thus transposes the adjective and the pronoun in a phrase.
_{The Shakespeare Key}

The book gives many examples of this, including "dear my brother", "dear my lord", "gentle my lord", and "good my glass".

The famous work A Shakespearean Grammar (1870) by E. A. Abbott (online here and elsewhere) should be consulted, even if it does not give the final word. Abbott considers that my lord is indeed at least treated as if it were one word, and likens it to the French monsieur.

On the transposition of possessive adjectives (which I suggest you read) he writes

The possessive Adjectives, when unemphatic, are sometimes transposed, being really combined with nouns (like the French monsieur, milord)

and proffers many examples, including

dear my lord

good my brother

sweet my mother

oh! Poor our sex

good your highness

good my knave

good my friends

good my mouse of virtue

Abbott writes

It is possible that this use of "my," "our," &c. may be in part explained from their derivation, since they were originally not adjectives, but the possessive cases of pronouns.

Thus, "sweet my mother," = "sweet mother of me," or "sweet mother mine."