We had this issue, but had problems figuring out where / how to submit this without a support plan. Eventually we received the following instructions from Amazon:

They stated the request could be submitted at the following URL: https://aws.amazon.com/premiumsupport/knowledge-center/ec2-port-25-throttle/

And to do the following:

1. Sign in with your AWS account root user credentials, and then open the Request to Remove Email Sending Limitations form.
2. In the Use Case Description field, provide a description of your use case.
3. (Optional) Provide the AWS-owned Elastic IP addresses that you use to send outbound email, as well as any reverse DNS records that AWS needs to associate with the Elastic IP addresses. AWS will use this information to help reduce the chance that email sent from the Elastic IP addresses is marked as spam.
4. Choose Submit.

For anyone who might have the same issue, you can submit a request to AWS support. Amazon only states that this is for EC2 instances, however it does the same thing for LightSail. I managed to resolve my issue this way. It takes up to 48 hours though.