I would be inclined to believe BA's figures, as they will be based on verifiable checkin data, reduced by those who are actually sick. The offer to ACAS to verify shows a high level of confidence.
BASSA don't even know how many members they have, crew have been too scared to communicate with BASSA that they are resigning and are simply stopping their payroll deductions. This is why many received ballot papers when they shouldn't have.
They may have received 7000 claims for strike pay, but there will be many multiple claims for different periods.
Can anyone produce verifiable rather than anecdotal evidence that BA have lied during this dispute?