Ah, so that's it

After a good four months' interval (not that it marks a break – swan-like (cygnesque?) my legs were thrashing about below the surface) I now have something new to show: the Notes for the /ɑ:/ bit of the AR section

1. applecart
   This escapes the usual exclusion for compound words, because the word is only ever used in the collocation "upset the applecart" and is not necessarily a cart for apples (or, indeed, any kind of cart).
2. archaeopteryx
   The Macmillan English Dictionary does not have the American English spelling.
3. archangel
   When "arch-" is used as a prefix (in English - so archvillain, archrival etc. but not archetype), the letters "ch" usually represent /tʃ/. In this case, though, the consonant sound is /k/.
4. archway
   "Arch", in this case, is not a prefix - if that were the case , one could expect to find statements like *"The M1 is the archway to drive from London to Edinburgh.
5. armchair
   This escapes the usual exclusions of compound words, because it is not a chair for arms - as in, for example, armrest.
6. barbarian, barbaric, barbarism , and barbarous
   This sound is in the first syllable. See also under /eə/, /æ/, and /ə/.
7. cardio-
   This prefix is used in many medical and physiological terms.
8. cartwheel
   This escapes the usual exclusion of compound wordshttps://harmlessdrudgery.blogspot.com/2012/11/but-nobody-says-potahto.html because, in its most common (metaphorical) use, it refers to an acrobatic movement that involves neither a cart nor a wheel.
9. compartmentalize
   The Macmillan English Dictionary does not have either compartmental or compartmentalise – though it does have compartmentalize with an audio sample marked as "British" (not that -ize spellings are necessarily unBritish [see this post for more details]).
10. farthing
    Students of English for Speakers of Other Languages should note that unlike in other nouns called "-thing" the fricative is voiced.
11. hardball
    This escapes the usual exclusion of compounds because (in British English) it is only used in the collocation "play hardball" (borrowed from American English).
12. hardline
    This escapes the usual exclusion of compounds because it is used metaphorically (to mean "strict").
13. lodestar
    This escapes the usual exclusion of compound words because the fossil "lode" that appears in it has no current life as a free-standing word.
14. margarine and margarita
    The first syllable has this sound. See also under /ə/.
15. marshmallow
    This escapes the usual exclusion of compound words because in it the word mallow has no relevance to the sweetmeat.
16. narc
    An American English usage, not to be confused with nark (who is on the opposite side of the law).
17. parliamentarian and parliamentary
    This sound is in the first syllable. See also /eə/.
18. pockmark
    This escapes the general exclusion of compounds, as the word "pock" is rarely if ever used as a standalone word. And when it is, it is probably understood as a back-formation from the "compound".
19. [See PS]
20. sarsaparilla
    This sound is in the first syllable. See also /ə/.
21. sidebar
    This escapes the usual exclusion for "-bar" compounds because of its metaphorical use to mean a separate channel of communication.
22. tartar
    This sound is in the first syllable. See also under /ə/.
23. Tartare Sauce
    This sound occurs in both syllables.
24. unparliamentary
    This sound is in the second syllable. See also under /ə/.

PS – I initially discounted quark as too specialized, but this thought is quite pleasing.

1. quark
   The Macmillan English Dictionary lists only the sub-atomic particle, and does not give the alternative pronunciation (with /ɔ:/). This may reflect the fact that when James Joyce used it in Finnegan's Wake the context suggested to Gell-Mann (namer of quarks) that it should rhyme with the name "Mark". However, apart from a name for a kind of cheese, this word is the only English word  with the sound /wɑ:/ represented by  the "ar" spelling; in fact, the sound /wɑ:/ represented by any spelling, is not very common (discounting the dialectal twa, a few foreign borrowings such as bwana and suave, and the comic-book conventional representation of an unhappy baby – wah!) (discounting the dialectal twa and a few foreign borrowings such as bwana) . This uniqueness might explain some peoples' (sub-conscious) preference for the alternative pronunciation.

Update: 2018.03.13.16:00 – Added to PS in red.