Just focusing on the $\cos^l\theta$ term is probably not going to get you anywhere, since $\cos^l\theta$, being a completely analytical function, is by no means singular (and following your argument you get exponential growth of $U_l$, not decay). It is a fairly well-known fact that for analytical functions (this is what "regular" means, roughly speaking the function is equal to its Taylor expansion. And here, we require analyticity in a finite region on the complex plane), the expansion coefficient with Legendre polynomial (also for Chebyshev, etc) decays exponentially. The proof is not that trivial, as you can see from the requirement of complex analyticity it uses contour integral. You can find the proof in many textbooks, for example

Philip J. Davis, Interpolation and Approximation, Dover, 1975

The proof for Legendre polynomials is at page 313.

Just focusing on the $\cos^l\theta$ term is probably not going to get you anywhere, since $\cos^l\theta$, being a completely analytical function, is by no means singular (and following your argument you get exponential growth of $U_l$, not decay). It is a fairly well-known fact that for analytical functions (this is what "regular" means, roughly speaking the function is equal to its Taylor expansion. It is a stronger condition than being infinitely differentiable. And here, we require analyticity in a finite region on the complex plane), the expansion coefficient with Legendre polynomial (also for Chebyshev, etc) decays exponentially. The proof is not that trivial, as you can see from the requirement of complex analyticity it uses contour integral. You can find the proof in many textbooks, for example

Philip J. Davis, Interpolation and Approximation, Dover, 1975

The proof for Legendre polynomials is at page 313.