Does FOL extended with least-fixed points satisfy the Compactness Theorem?
No, it does not satisfy compactness anymore.
Least fixed point operators can let us define, for example, the standard part of a model of $\mathsf{PA}$. The standard part of a model $\mathfrak{A}=(A;+^\mathfrak{A},\times^\mathfrak{A},0^\mathfrak{A},1^\mathfrak{A})$ of $\mathsf{PA}$ is just the smallest subset of $A$ containing $0^\mathfrak{A}$ and closed under $n\mapsto n+^\mathfrak{A}1^\mathfrak{A}$. But this is exactly a least fixed point definition: the standard part of $\mathfrak{A}$ is the least fixed point of the operator $$\Phi(P): x=0\vee\exists y(P(y)\wedge y+1=x).$$
This in turn lets us pin down the standard model of $\mathsf{PA}$ up to isomorphism by the theory $\mathsf{PA}$ + &quot;Every element is standard.&quot; But this contradicts compactness: no compact logic can ever pin down an infinite structure up to isomorphism (this is really just the observation that the upwards Lowenheim-Skolem theorem is just a compactness corollary).