I am sure you agree that we have established there’s something odd about the way حافظ uses it and I am sure that is the whole point of raising the thread,
What has been established is that مستانه can mean ‘intoxicating’ if and only if it is used with باده with ezâfé/izafat.
I suggest a useful thread topic and I’m not being facetious:
“Do we know any word in any language that has a meaning different to its grammatical one, when it is used in combination with one other unique word?”
PersoLatin, I shall begin with a premise that my knowledge of the Persian language is minuscule compared with yours and other Persian speakers'. Having said that, I do not find Hafiz's use of مستانه for بادہ as odd at all. Let's take a look at the word مستانه.
Hayyim defines it as an adjective-Adverb from which I presume he is saying that it can be used as an adjective and an adverb.
مستانه (mastaneh) Adverb-Adjective 1. Like a drunkard. 2. Drunken, befitting a drunkard. Ex. نگاه مستانه a glance like that of a drunkard; a languishing glance.
The example that he provides is clearly adjectival as the word is defining a noun "nigaah". What kind of "nigaah"? ..."like that of a drunkard".
This is how Dehkhoda defines this word and provides several examples, all on the pattern of
بادۂ مستانه . Apparently, "Dutch courage" is translated in persian as "شجاعتِ
مستانه. It is obviously based on the adjective مست drunk/drunken.
مستانه. [ م َ ن َ / ن ِ ] (ص نسبی، ق مرکب ) چیزی که حرکات و سکنات آن بطور مستان باشد چون لغزش
مستانه و رفتار
مستانه و گریه ٔ
مستانه و جلوه ٔ
مستانه۔
Here is another example (but this time شراب instead of بادہ) from
Shāh Ni'matullāh Wali, whose mausoleum is in Mahan, a beautiful place close to Kerman.
همچو خُمِ شرابِ مستانه
گرم شو خوش بجوش خوش می باش
ؒ
شاه نعمتالله ولی
The same poet, here is using it as an adverb.
مستیم ز شراب وحدت عشق
مستانه سرود می سرائیم
An adverbial example from Rumi
چو باده بر سر باده خوریم از گلرخ ساده
بیا تا چون گل و لاله در آمیزیم مستانه
In Urdu's literary tradition, there are critics who would tear the poet to pieces (so to speak) if there were errors of grammar, idiom and usage. I am sure, the same must be the case within Persian literature. Critics and fellow poets would say that Hafiz's use of
بادۂ مستانه is incorrect. Are you or other forum members aware of such criticism?
I know you would n't be facetious towards me. By now I believe you know me too well!

I don't know *all* the languages, so it is not possible to comment. I assume you are stating that بادہ does not go with مستانه since مستانه here would mean مستی آور and بادہ itself is مستی آور! Well,
Shāh Ni'matullāh Wali, a contemporary of Hafiz also does not feel there is anything wrong with this usage.