I think you've made a bit of a mountain out of a molehill here. You asked BA if they could change it for you, they said yes, but asked to see some documentary proof.

I really don't see anything wrong with BA asking you to provide some evidence that your request to change the DOB was valid (after all it's not something that normally needs changing!). It is probably also due to the fact that such data could be used as a challenge question to maintain security of your account.

Without access to a fax machine, could you not have just photocopied your passport, popped it in an envelope with a letter asking for the change to be made? Job done. Personally, I don't think it's BA who is being "ridiculous" here.