Simplest way to receive email to a lambda - AWS Simple Email

I am new to SES and thought it would be a quick setup but its looking a bit more involved than expected. I also have to go through a client so I want to make sure I understand my options before I make any requests.

So I have a client who has set up an email mailbox in outlook, dev@clientdomain.com. I want to have emails sent to this go into a lambda for processing, so I was looking at SES. Do I need to go through the same setup as if I was going to be sending emails? That is do I need to verify a domain identity or can I get away with verifying just an email? Do I need to ask the client to publish an MX record on their domain?

Tbh I was hoping I could just some email address in AWS that is linked to a lambda, and then have the client email forward to this. Is there nothing straight forward like this in AWS? I am able to send emails with the client email following this, but how can I receive an email and parse it with a lambda the easiest way?

AWS Workmail offers exactly this functionality (email on hosted test domain that routes to lambda of choice)