I am not quite familiar with the use of "being". Do I get them right in the following example?
If "That being said" = "That having been said", then is it "his hair being a mess"= "his hair have been a mess"?
Maybo, I think it's important for you to realize that the construction you have asked about in this thread (the construction found in "his hair a mess") is used only in literary English. There probably isn't a single ESL class the world over which covers the construction.
"That being said" works a bit differently, in that "said" is not a noun phrase but a verb phrase headed by a past participle (or an adjective, depending on whether one finds it equivalent to "That having been said"). I didn't mean to sidetrack the thread by bringing it up.
The relevance of that other construction is simply that, like the one you have asked about, it is an absolute construction, from which "being" can be omitted. Similarly, "
Lunch being over, they went back to work" can be shortened to "
Lunch over, they went back to work."
It has occurred to me that the construction found in "his hair a mess" admits another type of paraphrase: "
with his hair
as a mess." Notice that whether it is paraphrased that way or as "(
with) his hair
being a mess," there is a subject-predicate relationship between "his hair" and "a mess."
Incidentally, the "with . . . as . . ." paraphrase doesn't work so well on the second verbless absolute construction in your sentence. Instead of
(?) "
with his only accessory as a brown bag," it would be better to reverse the order: "
with a brown bag as his only accessory."
Coincidentally, in researching the American poet John Greenleaf Whittier (1807-1892) on Wikipedia this morning, I came upon the following excerpt from a hymn he wrote, in which the type of construction you have asked about is again used twice in a row, but without "and" coming in the middle:
"O Brother Man, fold to thy heart thy brother:
Where pity dwells, the peace of God is there;
To worship rightly is to love each other,
Each smile a hymn, each kindly deed a prayer."
- John Greenleaf Whittier
I'd sooner use the "with . . . as . . ." paraphrase there ("with each smile as a hymn, and with each kindly deed as a prayer") than the ". . . being . . ." paraphrase ("each smile being a hymn, and each kindly deed being a prayer"), though both capture the meaning of the construction.
Sorry for the delay in my reply. I have enjoyed this opportunity to write about the construction, which is one of my favorites. In all my years on grammar forums, you are the first learner whom I have seen ask about it.