I've finally somehow noticed this, after searching for several weeks (and pissing off some of the moderators on another SE site).
According to this, the correct character is
U+02BC ʼ MODIFIER LETTER APOSTROPHE:
It's description makes that really clear, as it specifically mentions transliteration but non-specific to any (human or presumably other) language:
[...] is preferred where the apostrophe is to represent a modifier letter
(for example, in transliterations to indicate a glottal stop). In the
latter case, it is also referred to as a letter apostrophe. The letter
apostrophe may be used, for example, in transliterations to represent
the Arabic glottal stop (hamza) or the Cyrillic "soft sign", or in
some orthographies such as cʼh of Breton, where this combination is an
independent trigraph. ICANN considers this the proper character
for Ukrainian apostrophe within IDNs. This character is rendered
identically to U+2019 in the Unicode code charts, and the standard
cautions that one should never assume this code is used in any
(Emphasis mine.) Note that it mentions that the glottal stop is only one possible use, leaving this open to pretty much any conceivable phonemic/pronunciation.
This is in contrast to
U+2019 ’ RIGHT SINGLE QUOTATION MARK:
is preferred where the character is to represent a punctuation mark,
as for contractions: "we’ve", and the code is also referred to as a
punctuation apostrophe. The closing single quote and the
apostrophe were unified in Unicode 2.1 "to correct problems in the
mapping tables from Windows and Macintosh code pages." This can
make searching text more difficult as quotes and apostrophes cannot be
distinguished without context.
(Emphasis mine.) This one is, of course, only truly proper for punctuation, which can be ruled out given that these names and nouns always seem to represent singular words without any indication that they are contractions (in the few extant examples of multi-word usage, such as Lovecraft's famous gobbledygook cultist chant, these are merely multiple words that are using it as a modifier letter of some type).
This late, I did not expect a definitive answer to exist, and certainly if one did exist either that I would be the one to discover it or that it could be anything other than "Unicode has no proper character for this. I've been through a bazillion blog articles describing the various differences of the apostrophe-like characters, and I must have passed this one by a dozen times reading through Unicode technical documentation.
I had gone so far as to have started a draft proposal to send in to the UTC, had it 80% complete only to stumble onto this while writing a section idiotically titled "Anticipated Objections". Ugh.
My proposal would have unceremoniously been rejected, and knowing what I know now, I don't think I could make even a bad faith argument that another character is warranted for this particular use. For the record, it was going to be UNSPECIFIED ALIEN PHONEMIC SYMBOL, in Latin Extended-F.
I'm going to let Wolff's answer remain the accepted (he answered with a well-considered and well-researched answer given in good faith, I appreciate that) and leave a comment under it directing anyone to come look at mine as well. Thanks everyone who answered/commented.