The objection from BHX is to do with the cranes rather than the height of the building.
I'm guessing there's a legal requirement for ATC to be made aware of tall cranes, otherwise the airport may be held responsible if we were to have a repeat of the London helicopter crash:
London helicopter crash: Two die in Vauxhall crane accident
Two people are killed and 12 are injured when a helicopter crashes into a crane at the top of a building in Vauxhall in central London.www.bbc.co.uk
Soon BHX will have to get there heads around the fact that we are also proposing a tower on Broad street which will be 193 metres, oh and we need a crane on top to build the thing. Can you imagine there faces ?