Originally Posted by Globaliser

The next (and possibly final) level of geekiness on the phone to BA would be to check that the fare that they're seeing is INNT00S7, the base fare of which is £513.50 each way or £1,027.00 round-trip for the adult. (ITA will add £10.50 each way to these figures.)

The companion fare is INNT00Y7, which would be at £338.50 each-way or £677.00 round-trip for the adult. You could ask the agent to try to force a price quote for adult + lap infant using this precise fare basis code.

My experience of doing this is when the auto quote starts failing a few hours before the end of a sale, but the call centre agent can still get the sale fare by specifically asking for a quote using the sale fare basis code.