Let's consider CS functional for concreteness. The problem is that CS is neither Morse nor Morse-Bott (because its critical points are flat connections and character varieties of 3-manifold groups could be rather bad). The trick is to perturb CS to a Morse function. This was done first by Taubes ("Casson's invariant and gauge theory") and then developed into Floer theory.

Let's consider CS functional for concreteness. The problem is that CS is neither Morse nor Morse-Bott (because its critical points are flat connections and character varieties of 3-manifold groups could be rather bad). The trick is to perturb CS to a Morse function. This was done first by Taubes ("Casson's invariant and gauge theory") and then developed into Floer theory.

How far is CS from being Morse-Bott? If you consider $SU(2)$ connections then for Seifert manifolds the character variety can have quadratic singularities, so it's not a manifold. There are examples of hyperbolic manifolds so that the $SU(2)$ character variety has cubic singularities. In fact, for $SO(3)$ flat connections over 3-manifolds the situation is much worse and you can have any singularity over ${\mathbb Z}$. I suspect the same happens even in $SU(2)$ case but it's harder to prove. So, perturbation to a Morse function is the only way to go.