Can I give some advice as an insurance broker. It all depends on whether you can establish your neighbour's actions were negligent or not.
If you can prove (or have the neighbour's insurance accept) that your neighbour acted carelessly in where he left the bin or HOW he left the bin, then you have a claim against your neighbour.
ALL home buildings insurance has public liability cover as standard and that's the part that would pay your claim.
Of course they don't want to deal with it but I would insist (and quickly) that they must.
The way you do this is by writing to your neighbour setting out your grounds for holding him responsible.....but hand it to him with a nice explanation of the process and a request he immediately submits a claim to his insurer.
On no account would I agree to use my own motor policy except as a last resort. If you do use your own motor insurance they will probably not pay unless you gain their approval/inspection in advance of the repair being done.
Final bit of advice therefore is to contain your impatience and CANCEL the repair until you see who is going to pick this up.....may take some time especially as your neighbour's house insurance may drag their heels.
You should also be aware that damage caused by bad weather which could not reasonably have been foreseen in cause or effect may absolve your neighbour from any liability.
This is where a good friendly insurance broker really comes into their own and direct insurers do not.

Good luck and please ask more questions if you need.