  In [arXiv:1906.10100], the authors argued that the Newman-Janis algorithm on
the space of classical solutions in general relativity and electromagnetism
could be used in the space of scattering amplitudes to map an amplitude with
external scalar states to an amplitude associated to the scattering of
"infinite spin particles". The minimal coupling of these particles to the
gravitational or Maxwell field is equivalent to the classical coupling of the
Kerr black hole with linearized gravity or the so-called $\sqrt{\text{Kerr}}$
charged state with the electromagnetic field. The action of the Newman-Janis
mapping on scattering amplitudes was then used to compute the linear impulse at
first post-Minkowskian (1PM) order, via the Kosower, Maybee, O'Connell (KMOC)
formalism. In this paper, we continue with the idea of using the Newman-Janis
mapping on the space of scalar QED amplitudes to compute classical observables
such as the radiative gauge field and the angular impulse. We show that for
tree-level amplitudes, the Newman-Janis action can be reinterpreted as a
dressing of the photon propagator. This turns out to be an efficient way to
compute these classical observables. Along the way, we highlight a subtlety
that arises in proving the conservation of angular momentum for scalar
-$\sqrt{\text{Kerr}}$ scattering.
