logic suggests "lawyer" should sound like /lɑjɚ/, as "lawyer" is basically "law" + "yer"
Since when is English pronunciation logical? As Wiktionary notes, outside of the South lawyer is pronounced in American English as /ˈlɔɪ.ɚ/ (phonemically). Liar, meanwhile, is transcribed by Wiktionary as /ˈlaɪ.ɚ/; some dictionaries don't list lier, but Wiktionary transcribes it as a homophone of liar. The diphthongs /ɔɪ/ and /aɪ/ are not affected by the cot-caught merger. Wiki doesn't list, and I'm not aware, of any American accents that merge /ɔɪ/ and /aɪ/, so outside of the South these two are certainly not homophones.
In the Southern US, per Wiktionary, lawyer can be pronounced /ˈlɔ.jɚ/. Such accents generally lack the cot-caught merger, though they may have the rod-ride merger (see Wikipedia), which would merge /ɑ/ and /aɪ/. Hypothetically, if a speaker did have both the rod-ride merger and the cot-caught merger, then lawyer (/ˈlɔ.jɚ/) would indeed be a homophone of lier/liar (/ˈlaɪ.ɚ/); the presence of the /j/ isn't particularly relevant, since the diphthong /aɪ/ would end in a glide anyway. But I suspect that such speakers are fairly rare, since accents with one merger generally lack the other.
There is also a merger between /aɪər/–/ɑr/ in some Southern AmE dialects; this can merge tire and tar, but not lier/liar and lawyer, since the latter has two syllables.
(As a general reminder: even in speakers that, as you say, realize the American /ɑ/ as a central [ä], we would still transcribe that phoneme as /ɑ/; slashes are for phonemes and square brackets are for phones. I think your question may have gotten this distinction confused.)