If claiming through your insurance, you are bound by contract. If you have an excess, you will have to pay it. Your insurer should try to recover their loss and you should use your legal assistance (if you have it) to recover yours. The issue lies in the cyclist not being required to have insurance. This does not mean that you cannot make a claim from them though. This being a civil matter, I would contact the cyclist and advise that you will be issuing proceedings against them in the small claims court. To assist matters of the claim, you would like to request if they have any insurance cover in place by way of their home insurance or any other policy?
You may find that they do have cover in place, It is amazing what "pending litigation" uncovers. There is no reason your insurer cannot do this, they are just inherently lazy.