I'm not sure you will win your claim. You have already stated that they have offered you a reroute. The question of whether a 6 hour delay is a reasonable one is debateable but if it is the next BMI or LH flight I imagine it would be deamed to be. If they have options on their flights I would also giuess they wouldn't have to reroute you on their competitors.
All their options could be upto how they want to treat you including only going to LHR. But I don't think legally they have done anything wrong. You can argue whether morally or just for customer service if they should of offered move than they did. It willbe interesting to see what they say.