This is just MY opinion for MY child, but the day my kid needs a psychologist to do her optional sport activity is the day we start looking for something else to do. And I say this as the parent of a kid who has had fears and mental blocks for years over many things (BWO on beam, BHS on beam, twisting on floor, giants, Tsuk vaults, etc.)
If her anxiety and fear was creeping into other areas of her life and making her not be able to handle daily life & tasks, it would be different. But the fact that she was afraid to do a BHS in a skinny piece of wood 4 feet in the air sounds perfectly normal to me.
ETA: I’d have no problem with my child participating in a group thing at the gym, but I, personally, would not be signing her up to meet with someone individually over fears that are specific to gymnastics.