From personal experience I would avoid copying files between servers if possible as you are going to experience issues where your servers are out of sync.
Personally, I would always go for the single source of truth option. For my servers behind a load balancer I have a master server that exports the shared folder via NFS and the other servers mount the NFS share in read-only mode and removes any sync issues
It does introduce a single point of failure but it's fit for purpose for my needs (It's not for a laravel app but a config management system but logic holds true)
However, depending on how often the keys change, if ever, will determine your approach. If your environment isnt very dynamic then the easiest might be to just copy the keys manually to your server before introducing to the cluster. If you a dynamically scaling based on load then chances are you using a configuration management too like puppet so you could look to add you keys here and centrally manage that way (or use a central file location with NFS).
The one thing I would say is you need to ensure you auth keys are secure so if you decide to use something like Amazon(I would personally keep my keys inhouse) lock it down!