Hi Nicholas,
All our S3 buckets are in region "US Standard."
Bucket names are all lowercase and include one or two hyphens, usually in the format clientname-backups (e.g., "masource-backups").
"Use SSL" is enabled in the "Upload to Amazon S3" post-processing engine for both sites.
Of the two websites I refer to above, the first is rather large, with the backup totaling 2.78Gb, broken into two parts: 2.0GiB and 801.8 MiB. The second website is much smaller, with only one backup part totaling 107MiB.
I have another website with Akeeba Backup installed running version 4.3.1. I am currently able to backup to S3 just fine (without the 4.4.0 update). I suspect the upload to S3 would fail after the update to Akeeba Backup 4.4.0, as with my second website mentioned above. I'm willing to try that and provide you any output (screen shots, log files ... whatever you need) to help troubleshoot. Let me know what steps you need me to take, and what information I can provide.
Thanks,
-- Joey