Have parameters been determined for the Fourier components of a Milky Way spiral density-potential model?
Yes.
Antoja et al. (2011) created two models: The Tight-Winding Approximation model (TWA) and the PERLAS model. They decided to use just the $m=2$ term for their TWA, based on data from the Spitzer Space Telescope.
The amplitude $A(R)$ was
$$A(R)=A_{sp}Re^{-R/R_{\Sigma}}$$
Their $\phi_2(R)$ (denoted $g(R)$ in the paper) was of the form
$$g(R)=\left(\frac{2}{N\tan i}\right)\ln\left(1+\left(\frac{R}{R_{sp}}\right)^N\right)$$
where $N$ measures, in their words,

how sharply the change from a bar-like to spiral-like occurs in the inner regions.

They approximated using $N=100$, although $g(R)$ should properly use the limit $N\to\infty$. $i$ is the pitch angle of the arms, which they measured to be $15.5$ to $12.8^{\circ}$. $R_{sp}$, the beginning of the spiral, was found to be $2.6$ to $3.6\text{ kpc}$. For the other constants, $A_{sp}$ was taken to be $650$-$1100$ or $850$-$1300[\text{km s}^{-1}]^2\text{ kpc}^{-1}$, depending on the exact arm shape, and the scale length $R_{\Sigma}$ was set to $2.5\text{ kpc}$.
The pattern speed $\Omega_p$ was set in the range $15$-$30\text{ km/s}$, which fits nicely with other values I've seen. $\theta_0$ (denoted $\phi_0$ in the paper) was set to $88$ to $60^{\circ}$ - a little bigger than the spiral arm pitch angle. putting all this together, we get a potential of the form
$$\Phi_\text{spiral}(R,\theta,t)=A_{sp}Re^{-R/(2.5\text{ kpc})}\times\cos\left(2(\theta-88\text{ to }60)+\left(\frac{2}{100\tan (15.5\text{ to }12.8^{\circ})}\right)\ln\left(1+\left(\frac{R}{2.6\text{ to }3.6\text{ kpc}}\right)^{100}\right)-\Omega_{p}t\right)$$
A $z$-component could be multiplied to get a three-dimensional potential, likely exponentially decaying with height.
Griv et al. (2013) also used density-wave theory. Their potential was
$$\Phi(R,\theta,t)=\Phi_{\text{mean}}(R)+\tilde\Phi\sqrt{\frac{R_0}{R}}\cos\phi$$
where
$$\phi=\phi_0+m\left[\Omega_pt-\varphi+(1/\tan p)\ln(R/R_0)\right]$$
which is essentially the same potential as discussed in the question, with
$$A(R)=\tilde{\Phi}\sqrt{\frac{R_0}{R}}$$
The authors chose $m=1,2,3$ components, with different parameters for each. $p$ (in degrees) ranged from $-6$ to $-20$, while $\phi_0$ (also in degrees) ranged from $239$ to $247$. $\Omega_p$ was from $32.4$ to $37.6$ ($\text{ km/s}$), also consistent with other models. $\tilde{\Phi}$ saw the most variation, ranging from $-65$ to $-390$ (in $\text{kpc}^{-2}\text{s}^{-2}$). $R_0$ was set to $8\text{ kpc}$, I believe. Again, an exponentially decaying function should account for the $z$-dependence.