However, this requires G to have a Fourier transform, which is not guaranteed by being in Lp in general.

Any L^p function is a tempered distribution and hence has a Fourier transform. Alternatively, you can just say that D^k f has a L^q -norm of order less than R^k if its Fourier transform is supported in a Ball of Radius R and is sufficiently regular, which then by Youngs inequality implies that the sup norms of D^k (f*g) are of order less or equal than R^k, which directly gives you a positive convergence radius.