It is true to say when reversing, liability may rest with the driver if they then hit something. It sounds like the scooter came at you from the side angle and a collision occurred.
They should not be used on pavements (same for cyclists) and my view is your insurer's response is unreasonable.
Given the dispute with the insurer, you can avail of the Financial Ombudsman Service dispute resolution scheme - which is much quicker than court action and free to use. You have to go through a complaints procedure first.
You are covered under the Consumer Rights Act 2015 here which means you have a right to expect a service (from the insurer) carried out with reasonable care and skill. Furthermore, the policy should be “fit for purpose” and “as described”. If it isn’t then you can allege breach of your consumer rights and breach of contract.
You should now make a formal complaint and if that does not resolve the matter ask them for a "deadlock letter", which is a letter giving their final response. Their complaints details will be available on their website if you take a look. You should do this right away in my view.
Once you have the deadlock letter the next step would be to escalate this to the Ombudsman - they will investigate and liaise with the insurer which would hopefully result in a reinstatement of cover under the policy. They can order the insurer to make a financial award for inconvenience if they have acted poorly.
Once you have their final response, you can make the complaint to the Ombudsman here : www.financial-ombudsman.org.uk/consumers/how-to-complain
Or by email:***@******.***
Based on the circumstances, assuming the insurer does not uphold your complaint, I am sure the Ombudsman will do. The Ombudsman will look at this case independently and will make a decision based on what happened. Their decision is binding on the insurer.
If the Ombudsman did not find in your favour, therefore, you then have the option of suing the insurer for breach of your consumer rights and you have up to 6 years to bring a claim to the court (the limitation period is 6 years from your date of loss, to when you need to issue a claim). This is considered as a last resort though - I am sure the Ombudsman will resolve this given the facts. If you get to this stage you may well need a law firm as you would be saying an insurer who has deep pockets and will very likely defend the claim. Perhaps revisit this site in future if you need further details of suitable law firms as I can point you in the right direction.