Improper placement of a comma

Consider the following sentence, "Skye will become a doctor, and a priest if he finishes theology school."

My understanding is that the comma before the "and" does not follow the rules for comma placement. So that one must re-write the sentence to be:

Skye will become a doctor, and he will become a priest if he finishes theology school.

In the above sentence the "and" is connecting 2 independent clauses and therefore a comma before "and" is justified. Am I correct in saying that only the second sentence is grammatically correct in terms of comma placement?