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I know that through the GUI, its completely possible (https://confluence.atlassian.com/bamboo/moving-plans-to-a-different-project-289277189.html) but I can't see anyway through the REST API to accomplish this.
You are not able to find a way because there is no REST API endpoint for that in Bamboo. You will need to use the UI.
I've created a feature request to create an endpoint for this as you can see here: [BAM-20325] Create a REST API endpoint to move plans.
In case you want to know how we prioritize and decide what to implement from our backlog, please check Implementation of New Features Policy - Atlassian Documentation.