No, you can't move lightsail instances to another account directly.
Apparently you can move EC2 instances between accounts ... so, in theory, you could export your lightsail instance to EC2 and move that ... but then your instance would be in EC2 and not lightsail.
Good question. Typically, moving resources between accounts is tricky as accounts are used as a boundary between resources.
I would suggest trying to use tagging. Tagging is a good way of aggregating spend by tags - you could tag some resources with one project and the other resources with another project tag such as the key value pair of "project": "Project Alpha"
Lightsail supports tagging as well:
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