I have a website www.example.com and an api www.api.example.com
I'm requesting that my SES (emailer) usage limits be raised using this AWS form.
The from address in my emails is team@example.com but the emails are actually sent using SMTP from www.api.example.com
Which domain do I supply to Amazon for the static reverse DNS record?
You don't actually need to do that, if you just switch from using port 25 to using port 587.
If you want to do it anyway, it doesn't really matter for this purpose which hostname you use, as long as the hostname is valid -- meaning, it has an A record pointing back to that same IP address and not to any other IP addresses (a single A record can return multiple IP addresses, and you probably don't want that, here).
You might not want to tie it directly to your www addess, though, since you might move that to an ELB, later. You can (and probably should) create a new hostname and point it to the elastic IP, such as smtp.example.com, where example.com is your domain name.