Static IP for Serverless app

I want to use serverless to create an API. Obviously I will want to use my custom domain name for the api eg:

I notice that serverless has some tools and plugins that allow you to do this using route53, however for me this is not possible as there are some corporate restrictions that mean the domain name must be owned/managed by the company that are buying the API from me. That is they simply want me to provide them with a static IP for the endpoint and they will set up the DNS on their own servers.

So my question is this.

Can I create a serverless app (which uses AWS API Gateway for managing usage and API key) with a fixed IP address?

No you cannot. You might be able to have the company CNAME whatever domain they are planning to use to your custom domain name?

Yes, thanks, I think this may be the only solution.