I've been Googling and using translators for months and can't find a definitive answer, and seeing as I don't personally know any Germans, I'm hoping someone here might be able to help.

Short version:
Is "In mitternacht, die Mensch-Maschine" grammatically-correct?
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Long version:
I really want to get the German line in Ultravox's I Want To Be A Machine as a tattoo so it has to be EXACT... therein lies the problem...

The original releases don’t include lyrics.
A Japanese reissue does but excludes that line.
The CD remaster does: “Im mitternacht, die mensch maschine” which translates to “in midnight, the human machine”. That can't be right.
Their official site doesn’t have lyrics for the Foxx-era albums.
Lots of online sources have the first word as “in” and includes a hyphen, which changes the translation to “at midnight, the man-machine” which is much better, but then someone on YouTube thinks the first word is "um". Um...
Listening to the recording itself, I’m sure he says “in”, but also “the” instead of “die”, which I can just ignore as I don’t need it bilingual.
So, is that it - “in”, “die” and the hyphen?
But then checking the words separately, "at" translates to "beim" and "man" is "mann"!

Any help would be massively appreciated!

I just double-checked in the actual song, and I've studied both english and german (native german, MA level), so I know what I'm saying.
in the song itself, the guy sings "in" definitely, which is grammatically incorrect. you can also clearly hear that the guy is not a native german speaker.
now re your tattoo, I guess you'll have to decide yourself what to do about it. I'll give you the way it is in the song and the way it should be if it was "100%" proper german:

what he sings:
"In Mitternacht, the Mensch-Maschine"

what it would be if was kind of (more on that at the bottom) proper german:
"Um Mitternacht, die Mensch-Maschine"

"at midnight" = "um Mitternacht", but first words in a sentence or line are always capitalized in german, so it would turn into "Um Mitternacht".
the hyphen is correct, since it is probably a reference to kraftwerk anyway. I guess it would even be okay without the hyphen ("Menschmaschine"), but with the hyphen it underlines more markedly the difference between the two spheres or domains the words are taken from (nature/technology), as well as the artificial or synthetic character of the word.

HOWEVER: to a native german speaker, the original line sounds kind of awkward either way, especially with the english "the" thrown in there. the thing is: from the way it's worded in the actual song, it can never transition into a proper german sentence (has to do with the weird word order and the position of verbs in german). it's basically just two disjointed phrases thrown together that cannot be understood as proper, coherent clause constituents of a grammatically correct german sentence. the comma also points to this.
the only way I could imagine this might work would be as a parody by a native english speaker on german in general or kraftwerk in particular. but I'm not sure if this was ultravox's original intention, since I'm not familiar enough with them. my guess is that it's just an example of bad internal quality control or proof-reading/-listening.
I'm really crossing into nerdy linguistics territory here, so I'll leave it at that. make of that what you will, and feel free to PM me for further discussion on this.

Just a heads up, Die Mensch-Maschine is a reference to the Kraftwerk album of the same name, so many people would think the tattoo is a reference to Kraftwerk.

Oh wow, it gets even more complicated! Thanks, into_the_pit. I'm not sure now...
(I knew about the Kraftwerk connection, but thanks anyway Belastbar.)

Then I suggest you abandon that idea and instead get a tattoo saying "At midnight I want to be grammatically correct".

You could write and ask the band members.

"Beim" means "at the" and is a contraction of "bei dem", since "bei" is a dative preposition and so that the article doesn't have to be used to save time if the noun is masculine or neuter. "Bei" is used to indicate locations. But when referring to time, "um" (which in other contexts means "around" as well) translates to "at". "Um 12 Uhr" = at 12 o'clock. "Um Mitternacht" = at midnight. "Im Mitternacht" literally translates as "in the midnight", and sounds wrong in either language.

"Mensch" simply means a human in general. Equivalent to English "person", more or less. "Mann" specifically refers to a male human. However, "man" has historically been used as a neutral term within English to represent a human in general, so that's where that translation might come from.

The other person here did point out that the line is rather weird-sounding as written. I'm not entirely sure about that, but for a more accurate version of the idea in the line, a phrase like "die Mensch-Maschine um Mitternacht" could be used. "The human-machine at midnight." But even then, it's still not a sentence in either language. Perhaps it makes slightly more sense in English, but I'm not a native German speaker so I can't attest to its legibility to a native speaker.

Thanks everyone for the input.

So what does "in mitternatch" mean?

I wouldn't have had this problem if I had gotten this 20 years ago. It would've been wrong but I wouldn't have cared.

I explained already. It quite literally means "in midnight". Which sounds wrong in both languages.

if you do decide to get the tattoo the way it's written/pronounced in the song, make sure you never show it to a german when drunk, because you're going to get ridiculed badly for it.