There is no evidential value in breathalysing a cyclist, so there is no point doing it.
Of course there is. Whilst there may not be an attached power to require a specimen of breath or a statutory limit to work with, getting the cyclist to voluntarily give a breath sample could have significant evidential value down the line. If the cyclist was steaming drunk it may have gone some way to proving the OPs story.
You CAN be unfit through drink or drugs on a mechanically propelled vehicle (IE a pedal cycle) but this has to be evidenced by the officer through other means and generally is corroborated by a force doctor.
A pedal cycle is not a mechanically propelled vehicle. There is an offence under some section of the RTA that is specifically "riding a pedal cycle whilst unfit through drink or drugs" though. There is also one under the licencing act which is something like "being drunk in charge of a carriage in a public highway".

Op is does sound like a strange one and one I suspect is going to have to go to insurance to decide on how it gets resolved. I would think about how you would put it to them and also mention right from the off that in your opinion you have no doubts that she is trying to scam
Insurance money due to a) the bizarre nature of the entire incident and b) it was the first thing she started bleating on about when she went down.