Can an electromagnetic potential be in two gauges at once?

I have a plane wave vector potential found using the free field form of Maxwell's equations and the Lorentz gauge:

$\vec{A}(\vec{r},t) = \vec{A}_0 e^{i(\vec{k}\cdot\vec{r}-\omega t)}$

If I take the divergence of this, I get

$\nabla\cdot\vec{A}\propto\vec{k}\cdot\vec{A}_0$

If I choose $\vec{k}\cdot\vec{A}_0 = 0$ then I would have a vector potential in the Coulomb gauge that I derived by using a potential in the Lorentz gauge.

Is it okay to do this and have a potential in two gauges at once?

Yes, it is perfectly possible to satisfy more than one gauge condition. The easiest way to see that is to consider a static configuration of fields. In a static problem, all quantities are time independent. That means that the Coulomb gauge $\nabla\cdot\vec{A}=0$ and Lorenz (not "Lorentz"; it's named after a different physicist) gauge $\nabla\cdot\vec{A}+\frac{1}{c^{2}}\frac{\partial\Phi}{\partial t}=0$ are equivalent. In your example, the two are equivalent because there is no scalar potential $\Phi$. Note that your field is also a solution in temporal gauge ($\Phi=0$) and certainly others as well.