Well to be fair, you can create your own SMTP server somewhere else, it just costs money (and really any solution is going to cost something). I only mention it because I recently worked with a customer who went that route. But managing your own Postfix or Sendmail server just to configure it to strip out the received from header and forward it on can be a PITA if a. that’s all it is doing and b. you don’t speak SMTP for a living*.
I believe both Mimecast and Proofpoint might offer such a service for outbound mail. Generally services are reluctant to offer that because knowing the true origin IP address helps in fighting the good fight against Spam, but there are valid reasons such as this where one might want to strip that data.
If you decided to host your own server you could do something like the link below describes. But given your use case you could further lock down the server in EC2 settings to only allow SMTP connections from your origin IPs making it somewhat easier to manage/ keep secure.
And you could do something like https://posluns.com/guides/header-removal/ to remove the headers (I haven’t actually tested either link I just Googled for them).
*I used to speak SMTP for a living and I still think it’s a PITA so even that’s not a sure thing.
Hope that helps…