I am under the impression that a lot of people make the ɑː to aɪ change also in a word like Naomi (as naɪˈəʊmi). As a foreigner, I am unsure of that, since there are also a great many people who do not resort to a foreign value for the a and say neɪˈəʊmi.
The corpus of such words in everyday vocabulary is extremely low; but more data should be gathered by listening to the way anglophone media pronounce chains of vowels in Hawaiian and Polynesian proper names...
OK, let it be gathered. Who’s first?
Some people may be surprised to hear this, but the traditional RP form of Naomi has initial stress, ˈneɪə(ʊ)mi. You won’t find anything else in Daniel Jones. As far as I can see, Jack Windsor Lewis was the first to record the penultimate-stressed form, in his CPD (1972). So for traditional RP (and for me), Naomi does not illustrate the point at issue, which is the unstressed prevocalic vowel(s) spelt a.
There are, however, two Biblical names that are perhaps relevant: Canaan and Sinai. These words each have initial stress. For foreign words of this vintage, long a would be expected to be read as eɪ (as in Amos, Salem, Jacob, Emmaus), not as ɑː.
Here’s the twelfth edition of EPD (Daniel Jones’s last, 1963). Focus on the penultimate vowels Canaan, Sinai.
What we find in both of these cases is a tendency to reduce weak eɪ to i, and perhaps further to j or zero.
(Who would believe that Sinai used to rhyme with tiny?)