well you're technically solving a different problem in the second case, you won't always get the same answer and sometimes adding that can mess up your solver.

well you're technically solving a different problem in the second case, you won't always get the same answer and sometimes adding that can mess up your solver. However a very common method for handling tricky constraints is by "relaxing" them, by moving g(x) into the cost function, exactly as you propose. I think trajectory optimization can still be viewed as a point optimization, and it's helpful to understand the fundamentals before jumping into applications.