To clarify here, you said you "looked at the registered controller on my Composer HE". If you looked in account services, only the master controller gets "registered", and you would only see the HC-800 there.
If you were looking in the Monitoring tab in HE, you should see the slave HC300 there if it's in the project, but this isn't "registration".
Only 1 controller gets "registered". Every controller should show in the project. To answer the question, replacing the power supply wouldn't cause the need for any programming or re-identifying of the controller if the HC300 is in the project already.