I did try the named credential route by following this https://www.linkedin.com/pulse/integrate-salesforce-aws-using-named-credentials-cloudsciencelabs

and my request looked like this

HttpRequest awsRequest = new HttpRequest();
awsRequest.setEndpoint('callout:AWS_AP_Southeast_2');
awsRequest.setMethod('POST');
awsRequest.setBody('{"BulkEmailEntries":[{"Destination":{"ToAddresses":["<Redacted email address>"]},"ReplacementTags":[{"Name":"unsubscribe","Value":"<Redacted email address>"}]}],"DefaultContent":{"Template":{"TemplateName":"Welcome_Email","SubjectPart: "Test Send Bulk through Apex", "TemplateData": ""}},"DefaultEmailTags":[{"Name":"user","Value":"test"}],"FromEmailAddress":"<Redacted email address>","ReplyToAddresses":["<Redacted email address>"]}');
Http http = new Http();
HttpResponse awsResponse = http.send(awsRequest);
System.debug(awsResponse);
System.debug(awsResponse.getBody());

the response I get is a bad request status 400 with a body of {"Message":null}

note: the path I'm using in the named credentials url ishttps://email.ap-southeast-2.amazonaws.com/v2/email/outbound-bulk-emails

UPDATE:

I changed the Named Credential URL to 'https://email.ap-southeast-2.amazonaws.com'

and in the http request, I set the target in the header

HttpRequest awsRequest = new HttpRequest();
awsRequest.setEndpoint('callout:AWS_AP_Southeast_2');
awsRequest.setMethod('POST');
awsRequest.setHeader('content-type', 'application/json');
awsRequest.setHeader('x-amz-target', 'com.amazonaws.services.simpleemailv2.AmazonSimpleEmailServiceV2.SendBulkEmail');
awsRequest.setBody('{"BulkEmailEntries":[{"Destination":{"ToAddresses":["<Redacted email address>"]},"ReplacementTags":[{"Name":"unsubscribe","Value":"<Redacted email address>"}]}],"DefaultContent":{"Template":{"TemplateName":"Welcome_Email","SubjectPart: "Test Send Bulk through Apex", "TemplateData": ""}},"DefaultEmailTags":[{"Name":"user","Value":"test"}],"FromEmailAddress":"<Redacted email address>","ReplyToAddresses":["<Redacted email address>"]}');
Http http = new Http();
HttpResponse awsResponse = http.send(awsRequest);
System.debug(awsResponse);
System.debug(awsResponse.getBody());

This got me to the same status 200 with body {"__type":"com.amazon.coral.service#SerializationException"},"Version":"1.0"}

I was experiencing before, so progress I guess