If you go through your insurance, you'll generally have to jump through all the hoops of making a claim and they can treat you like a criminal.
I'd get in touch with Lidl and ask for details of their public liability insurance policy. They may well insist that you pursue the claim through your insurance company as agents, but try and deal with them directly. They're always nicer to you once they've accepted liability.
As with any dealings with insurance companies, try to get as much documentation and/or details as possible. Good luck!
