That reply is shocking!

I'd be tempted to push this one.
Say you're cycling in anyway. Ask them for the full H&S assessment, written of course. Ask them if they are aware of the stats (longer life, greater health vs risk of injury). Ask them if you are
required to drive to work, as walking is, per mile, more dangerous than cycling which they have decided is too dangerous. In which case, ask them for a car, petrol, insurance etc as they're imposing the restriction. Its only fair. In general, make yourself a pain in the nipsy.
Reading this, I'm lucky. I work for a diddy IT company, so I went up to the MD and showed him the scheme details. He okayed it there and then. We run it all internally. I already had a commuter bike at the time, but a couple of other employees have taken advantage and, guess what, its freed up some very restricted parking for visiting customers. I've since had to replace it after a crash and there was no restriction on cost or supplier. A happy Bollo.