Use Zephyr Scale Cloud API to add existing test cases to and newly created test cycle.

I'm not a user of the API, so can't verify this before suggesting, but to add test cases to a test cycle, I think you need to use the Create Test Execution endpoint, and add test cases with an execution status of Not Executed.  This would mimic what happens when adding test cases to test cycles in the UI, because what actually happens behind the scenes is that a test execution is created - in Zephyr, a test case can have multiple test executions, so technically, when you add a test case to any test cycle you are actually creating a test execution record for that test case.

I hope this helps - let me know if not?  And if you feel like posting your updated code when you have a solution that would be awesome for the community to have available!

Good luck.