Totally unreasonable. According to EU regulations, BA have to offer you a choice of a full refund or a rebooking on the same route at the earliest possibility or a day of your choice (when there is availability). They certainly don't have to offer to rebook you on another route, wouldn't have to do so in the case of a paid ticket let alone of an award ticket.
I think you will find that re-booking on the earliest possibility includes other airlines and that is what I would be asking for, departure the same day as the original flight, preferably within 2 hours of the STD. Might involve an indirect flight, but worth looking at.