Grammar check on the following, please.

Téigh leis an Bhanrion Mhor, a Dhuke, a mhadra is fearr

I was trying to say, "Go with the Great Queen, Duke, best dog"

How close am I?

Maybe

Téigh leis an mBanríon Mhór, a Dhuke, a mhadra is fearr.

But I'm not sure about the form for "mór".

Wait for more input.

Téigh leis an Bhanríon Mhór (so in Ulster; else: mBanríon Mhór), a Dhiúic, a mhadra is fearr.

"Duke" as a proper name is unchanged (a Duke).

Momentarily forgot that Banríon would need elipsis and not lenition in mosts part of Ireland (exception being Ulster where it would be lenition).
Wasn't quite sure about Mór. Thought it would require lenition regardless because it's modifying a feminine noun regardless of whether that noun has elipsis or lenition.
Had no idea at all about how to handle non Irish proper names when used in the vocative.

Would the general rule here be to use the proper name as is without lention unless there's an obvious Irish equivalent to the name? For example, "Hello, Mary" would be Dia duit, a Mháire but "Hello, Matilda" would be Dia duit, a Matilda. Am I correct on that?

Although I don't like to see it, I know many people who lenite English words/names used in Irish. I don't know if it is right or not, but it follows the grammar. Personally, I also wouldn't change the spelling of "Duke" using Irish orthography but that again is a preference. I would also eclipse "banríon".

How to pronounce "a Dhuke"?

There's rather a slender /d/ in English, so "a Yuke"?
Or, as orthography suggests, with a broad dh /​ɣ/?, "a Ghuke"?

No English speaking dog would understand

It wouldn't be far removed from "a Dhiúic", I imagine, as in the flow of speaking the pronunciation may come naturally. I get letters from people starting "A Thim, a chara . . ." and it sounds weird to me (Ah HIM) but again it is grammatical. I don't really know what is correct. To each his own I guess.