Does the word "pirate" use the /aɪɚ/ phoneme, or the /aɪɹ/ phoneme?

vowels rhotic

The Longman Pronunciation dictionary gives /paɪᵊr ət/ (also acceptable, /ɪt/), in both BrE and Gen Am. The word has two syllables and superscript schwa stands for the pronunciation of schwa, which is not recommended; this symbol means that occasionally the sound schwa is inserted and then the word contains three syllables: /paɪ ər ət/.

LPD, 2000 p. 757

Likely syllabic consonants are shown in LPD with the symbol ᵊ, thus suddenly 'sʌd ᵊn li. LPD's regular principle is that a raised symbol indicates a sound whose inclusion LPD does not recommend (see OPTIONAL SOUNDS). Hence this notation implies that LPD prefers bare n in the second syllable. Since there is no proper vowel in this syllable, the n must be syllabic. (Thus it is possible, though not usual, to say 'sʌd ən li rather than 'sʌd n li.

Fire

BrE: 'faɪ‿ə (The italic ɪ shows that the sound ɪ may be absent; one or two syllables; 'fa ə, 'faə)
GenAm: faɪ‿ᵊr (Schwa is replaced by syllabic r, but ɪ is always retained.)

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