Impossible to decompose EM plane wave in spherical waves? (Normalization mismatch)

I am having a problem with an expansion that should be simple. Let’s say I solve Maxwell equations in vacuum, but in spherical coordinates. The solutions of the TM family can be found easily to be
$$
E^{TM}_{lm}(k;mathbf{r}) propto sqrt{l}j_{l+1}(kr)mathbf{V}^m_l(theta,phi)-sqrt{l+1}j_{l-1}(kr)mathbf{W}^m_l(theta,phi),
$$
where $l=1,2,3,…$, $m=-l,…,l$, $k=omega/c$ is the wavevector, $j_l(z)$ is the spherical Bessel function of order $l$, and the angular dependence is given by combinations of vector spherical harmonics (not important for this post). A similar solution can be found for the TE modes.

Now, assume I have a plane wave $mathbf{E} = mathbf{E}_0e^{imathbf{k}cdotmathbf{r}}$ that I want to expand in terms of the above eigenmodes in spherical coordinates. More specifically, if the electric energy of such plane wave is $U = int dV epsilon_0vert mathbf{E}vert^2/2 = epsilon_0 vert mathbf{E}_0vert^2V/2$, I wonder how much of such energy lies in the spherical eigenmode $TM_{klm}$. Assuming the plane wave is decomposed as
$$
mathbf{E} = sum_{lm}sum_sigma c_{lm} E^{sigma}_{lm}(k;mathbf{r}),
$$
with $sigma = TM,TE$, we can find easily that the fraction of the energy going to the $TM_{klm}$ mode is
$$
F = frac{1}{vert mathbf{E}_0vert^2V}left(int dV mathbf{E_0}e^{-imathbf{k}mathbf{r}}cdot frac{E^{TM}_{lm}(k;mathbf{r})}{sqrt{int dV vert E^{TM}_{lm}(k;mathbf{r})vert^2}}right)^2.
$$
Now, the angular integrals are $r-$independent, and the radial integral dependence is easy to find as
$$
lim_{Rtoinfty}int_0^R dr r^2 j_lambda^2(kr)to frac{pi R}{2 k^2} propto V^{1/3},
$$
$$
lim_{Rtoinfty}int_0^R dr r^2 int dtheta dphi e^{-imathbf{k}mathbf{r}}cdot E^{TM}_{lm}(k;mathbf{r})propto lim_{Rtoinfty}int_0^R dr r^2 j_lambda^2(kr)to frac{pi R}{2 k^2} propto V^{1/3},
$$
for any $lambda$. Then, the fraction we seek is
$$
F propto V^{-frac{2}{3}} to 0
$$
in the limit of infinite volume.

This happens for all the modes, and is related to the fact that the mode normalization in Cartesian coordinates is proportional to $V$ whereas in spherical coordinates it is proportional to $V^{1/3}$. This makes me think that it is not possible to expand a Cartesian plane wave in spherical waves, but this has to be wrong based on physical arguments.

Does anyone know where am I making a mistake? or have I bumped into some real limitation of the spherical vs cartesian coordinates? If so, what is the physical intuition behind it being not possible to expand an electromagnetic plane wave in spherical waves?

Thanks!