0. Talk to the people asking the question and ask them what they are comfortable with. Relying on AWS lambda is great so long as your credit card still works, the code contains no bugs, there isn't a problem with the email, and somebody at the client's office understands what the email means and can quickly do what it takes to keep everything working...and there are no other problems or circumstances.
1. Consider the business value derived from continuing to be an information bottleneck. An alternative is to set up servers and databases on accounts controlled by your clients.
2. To me, the conversation suggests at a minimum the client has done a risk assessment. Though the questions may have been prompted by conversations with a larger competitor (a competitor only needs two programmers to provide a more robust bus factor).
3. This is a business problem not a technical problem. It is also a business opportunity. Maintaining servers and databases is commodity work. Expertise in solving the businesses problems and an ongoing relationship is value added work and can lead to better contracts and better clients.
Good luck.
An alternative, but similar thing, is to put your sources in escrow, so that they become available to the customers only in such an occurence.
Of course, you may also merely distribute it as free software, with the sources stored on multiple public git repositories.