Can anyone give me a grammatical explanation as to why "that being said" is proper English?

Solution 1:

Your pedant is completely wrong, not just because he's protesting in futility to a well-established idiom, but because his grammatical analysis of the construction is mistaken.

That being said is an adverbial participle phrase. Note that the verbal portion of the phrase, being said, does not contain a finite verb, and only finite verbs are tensed. (This is not just a fact of English, but a general facet of Western European verbal systems: non-finite verb forms generally indicate aspect and voice, but not tense.) The terminology is somewhat confused on this point since being is often called the "present participle", but there is in fact nothing that specifies the present tense about this construction, as the following examples show:

The lawn was being mowed yesterday.

The aquarium will be being cleaned all day tomorrow.

In all of these cases, the bolded portion does not contribute any tense information at all -- it rather indicates the progressive aspect and passive voice. The tense is entirely indicated by the finite verb in italics.

The idiomatic phrase that being said contains no finite verb and no tense. Your pedant has merely demonstrated his own ignorance of English grammar. There is absolutely no reason to object to this idiom.

Solution 2:

That being said, ... can be rewritten as After saying all that, ... .

In the statement said is acting as an adjective, so is equivalent to that statement being said, or analogous to that ball being bouncy. You could point at a ball and say that being bouncy, you can use it to entertain your dog.

It is fine to use said in an adjectival form in this sense, because it is describing the state of a statement; the state of having been spoken.

Said being used as an adjective is not unusual, even if it is chiefly used in law: http://dictionary.reference.com/browse/said - although this is a different type of usage to how said is used in "that being said".

That being said, your acquaintance is being a grammar maven, so can be ignored. ;)

Solution 3:

"That being said" I interpret as "That, now in a state of in the record of the current conversation, ..."

So it's similar to "That being a Ford..." or "That being green". "Said" is an adjective in this case.