The setup I was thinking of would be to host the website’s static files on S3, probably front-load that with Cloud Front CDN. You’d need to utilise Route 53 to setup the aname routing and Cloud Front layer too.
Then the dynamic bits: I’d create an API end point using API Gateway and using a php driven Lambda function to handle the cURL request.
If you’re more used to a typical hosting setup and the site is small then there’s not really any need to go for a setup like this - you may be more comfortable in spinning up a Lightsail server and putting that behind Cloud Front (or any other CDN) to handle the static elements in the same way while still routing to the php file as you want.
Utilising all of the AWS functions can be a bit of a burden for small projects. It’s fantastic for applications that require automatic scaling and reducing bare metal server management requirements, but it’s not for every project so carefully consider whether you need it this way would be my advice.