There is Moving from one S3 bucket to another, which is not very easy.
I think what I would do, if everything is still on the lightsail buckets as you say, is to
- turn on the hidden
SiteSetting.include_s3_uploads_in_backups=true
on the site with old buckets - set read only
- make a backup
- restore that backup to the newly configured system
- switch DNS to new system
I would keep the old system in place for a bit until you’re sure everything is straight on the new one.
If you have things in multiple buckets on a single instance things get much more complicated. I’ve got a couple of sites that way that I’ve been trying off-and-on to get straight for longer that I care to admit publicly.