electromagnetism - EM Fields in a Rotating Frame of Reference

I'm struggling on my approach to the problem of figuring out E and B fields in a non-relativistic way for a rotating frame of reference in the x-y plane around the z-axis. I am attempting to do this without the use of Tensors:

Consider a particle at rest with respect to the rotating frame. Since there is no velocity there is no force felt from $\vec{B'}$. Do I have to include the pseudo forces in this equation?

$$\vec{F} = m \vec{a} = q \vec{E'} = q(\vec E+\vec v \times \vec B) \\ \vec{E'} = \vec E+\vec v \times \vec B$$

Additionally, considering a stationary particle in the inertial frame of reference, with pseudo forces included:

$$\vec{F} = m \vec a + m \vec \omega \times \vec v + m \vec \omega \times \vec v + m\vec \omega \times \vec \omega \times \vec x \\ = q(\vec{E'}+\vec v \times \vec{B'}) = q\vec E$$

However if I just plug in $\vec{E'}$ into the second equation, $\vec{B'}$ = $\vec{B}$ which seems incorrect. Looking at the non-relativistic portions of a Lorentz transformations has $\vec{B'}$ = $\vec{B} - \vec v \times \vec{E}$