Provision Lightsail for a different account using a snapshot

OK, maybe an odd question :)

I have an AWS Organisation that I use for managing billing for different projects/clients. To make this happen, I create a new AWS account per client and add them to my Organisation. I then use the access credentials for the new account to build a Lightsail server in that account, and I configure that server and install my software using a launch script that pulls various components from my distribution server and the usual Ubuntu repositories.

This is OK(-ish) but not 100% predictable. What I'd really like to do is create a template Lightsail server with all the software installed and tested, make a snapshot of that server and then create the new client Lightsail instances from the snapshot.

I can successfully create instances from snapshots like this in the same account but it doesn't seem possible to build a snapshot in one AWS account and then use it to create another Lightsail instance in a different account, even if both are in the same organisation.

Is this correct, or have I missed something?

All help gratefully appreciated :)

David