Have you considered writing FB and explaining:
- the problem of getting to the airport to book the tickets
- your understanding that they try to prevent fraud by brokers
- your understanding why your behaviour may look like that of a broker
- proof that you are booking for your brother's family to allow them to join you on a family holiday
- with the request to allow you to book the flight or refund the miles?

Regarding your question of why they are against brokers, a hint: the number of award seats is not equal to the number of seats they think they will not sell, and FB miles are not fully "commodified". If this were the case, there would not be award tables, but instead each mile would have a clear monetary value for unlimited buying and a clear monetary redemption value that you could use towards the price of any seat available on a plane.