Just log into your Avios.com account and go to “Manage my Account” followed by “Combine my Avios”. There you can move Avios back and forth between Avios, BA and IB accounts.

The only constraint is that the account details on each account much match exactly (Name, DOB, address etc.)

I suspect the BA person you spoke to was confused by the fact that Avios accounts not linked to a Frequent flyer programmer are no longer available. Many in the UK would have had both but everyone got rolled into BA Avios a couple of years ago.