I did end up finding a solution for anyone who wants to serve S3 files (or any files for that matter) on a custom domain over HTTPS.
I setup CloudFront with an origin pointing to the S3 bucket, and then added a CNAME in my DNS provider that points to CloudFront. I also had to create a SSL certificate using Amazon’s free certificate provider. Now I get all my files hosted at “bucket.domain.com/filename” instead of “s3.us-east…”.