Have you spoken to them rather than just sending emails with bug lists backwards and forwards? I find it can sometimes resolve issues and misunderstandings much

Have you spoken to them rather than just sending emails with bug lists backwards and forwards? I find it can sometimes resolve issues and misunderstandings much quicker. I would speak to them and explain that the issues investigated so far are out of scope, why and how they can identify for future issues. Let them know that you will need to bill for time on out of scope work but that you will waive the fees to this point as a goodwill gesture (let them know the time/cost of time spent so far). Send them the list of bugs with you findings so far with an invoice totalling zero after discount and ask them to let you know which issues they still want you to investigate. Is there further business that you could offer them in fixing or replacing the other (broken) components.