I am sure, most of you must be aware of the female name ending bāī /bāyī/ bāy /bā found in many contemporary northern particularly north-western languages.
Pingree's version is possibly bāī >vāī, such a phonetic change b<>v being quite common many Indian languages.
The bhāī version could possibly be a hyperstandardization or scribal error.
The town "Vāī" (वाई) is found in Maharashtra, a famous place near Satara, where many Sanskrit works like Dharmakośa were published. Vai has been a well known center for Sanskrit scholarship and the home of the Prājña Pāṭhaśālā, which continues to function till today.