The technology to voyage over the south pacific definitely did not exists in 1600 BCE. So trans pacific human contact is an unlikely explanation. More likely these particular species might have migrated them selfs (e.g. by floating) or with non-human animals (like birds) and later picked up while still alive by human farmers in India.

There is also the possibility of more crossings over the Bering straight. Inuits and proto-inuit cultures have crossed the Bering multiple times. I find it dubious that they never crossed westwards. However, the technological niche which the cultures of the far north have mastered over the centuries have less values in warmer climates so I don’t think much would have been brought from the Americas to as far south as India via the Bering.

Anything crossing the south Pacific from around 500 AD however can definitely be attributed to human crossings as the Polynesian cultures from that time period for sure had the technology for such voyages.