Not sure it'd work on all crossbeaks and I definitely wouldn't tape if the bird acts like correcting the alignment hurts. This birds beak had only been crossed for a week and was from an injury. Suddenly occuring crossbeak. If the cross had been there longer or developed slowly I wouldn't have tried. Slow developing crossbeak means something's going on inside the bird that needs addressed.Wish I had know about this before I lost my most loved scissorbeak pulled a while back. If I had taped her like you did my may could have saved here. She had more out of alignment problems that here beak I think. Her keelbone didn't line up either. She may have had more problems internally not visible. She never laid an egg but was a pet like I had never had before. She loved to keep me company.
Sorry you lost your girl... sounds like she was special

Editing to add that a suddenly occuring crossbeak can get worse over time...so maybe the best qualifier would be to just say it probably won't work if the bird acts pained from the beak being held in proper alignment.