Of the origin of aubergine he says
This is, as you might guess, borrowed from French; the French word is from Catalan albergínia, which is from Arabic al-bādinjān (with the definite article al-), itself borrowed from Persian bādingān, which is probably from Middle Indo-Aryan *vātiñjana-, vātingana-; most sources attribute the latter form to Sanskrit, but I don't find it in my dictionaries.
Same here, altho I did find हस्तिवातिङ्गण hastivātiṅgaṇa in the Cologne Digital Sanskrit Lexicon.
The Oxford Hindi-English Dictionary says the source of बैंगन baiṃgan "eggplant" is Pali vātiṅgaṇa-, with a Dravidian origin. No other source mentions a Dravidian origin, so I don't know how certain that is. Tamil has வங்கம் vaṅkam, வார்த்தாகம் vārttākam and கத்திரி kattiri.