I think your friend has a point. I will not comment on whether this construction is acceptable; I will just explain why it is somewhat irregular. Consider this sentence:
Its walls being assaulted by the Crusaders, Constantinople got little sleep that night.
It is clear that the city's being assaulted does not precede the waking of its inhabitants: being assaulted happens at the same time as got little sleep. That is how present participles work: they happen at the same time as the main verb in the sentence of which they are a part. The absolute construction with being + past participle inherits (as JSBangs said) the time of got.
But assaulted is a past participle, you say: its action should precede the main verb in time! That is generally true, as in the following example:
The captured baron paid a ransom of two thousand marks.
Here the baron's being captured precedes his paying a ransom: he is not still in the process of being captured as he pays the ransom. But that does not apply to the Constantinople example. The reason for this is that the auxiliary verb to be changes the meaning of the past participle: usually, past participle + to be loses the sense of being earlier in time that is normally present in the past participle.
Back to the said example. You didn't provide a context, but based on your comparison with having been said I believe you were thinking of a sentence like this:
That being said, I will now proceed to a new topic: Borgia dominance in the Vatican.
If we assume that being + past participle happens at the same time as the verb of the main clause, and that I will proceed begins now, the "saying" of that being said should still be going on now. But I don't think this is what it is supposed to mean. That being said is here most probably used as a substitute for that said and that having been said: the speaker wants to tell us that the previous topic is finished. Then the phrase should be considered irregular.
However, perhaps we could take said as an adjective, not a past participle. Whenever a participle is used as an adjective, which means that it is not linked to an auxiliary verb like to be, it still has its sense of being earlier than the main verb, as in the captured baron above. But is this possible? There are some examples of this with other past participles:
Caesar, this catapult must be broken.
You could read this as Caesar, we must arrange for this catapult to be broken: then be broken is treated as auxiliary be + past participle. Or it could be Caesar, this catapult must be defective (how else do you explain why it failed to fire?): then broken is treated as an adjective and keeps its sense of having happened earlier; the catapult was broken some time ago and as a result it is now in a broken state. Note that be is a copula then, not an auxiliary verb.
Can we so interpret said in that being said? Then I'd paraphrase it as this being in a said state, I will now proceed. That is possible, but it certainly isn't the usual way to interpret said. Whereas broken is very frequently used as an adjective, probably much more often than as a true participle, this is not the case with said. Even so, I think this may very well be how those who use this construction subconsciously do it.
However, though this construction is quite common, I do not think it has any advantage over that said, which is both shorter and less convoluted. As far as style is concerned, I prefer that said.
If that being said is used to express an argument that has already been presented ("said") but is still being examined by the speaker, its sense of happening at the same time isn't a problem at all:
Most modern scholars say that Churchill was one of the few who had no sympathy for Hitler at all. That being said, it is no wonder that Chamberlain happily accepted Hitler's guarantee that he'd leave Poland alone.
Here that being said means considering that they say this (notice the present simple); the scholars never really stop saying this as long as we keep reading what they have written or using their arguments. This is similar to the timeless it says "he was a great poet" on his grave: it says so as long as the letters are still readable. This usage is not irregular at all. However, I don't think this is what the question was about, because it compared that being said to that having been said and that said, which indisputably have a sense of being earlier than the main verb. Substituting that having been said in the Churchill example would change its meaning completely.
