= Solution
<Relativistic beaming> concentrates the emission into a forward cone about the instantaneous <velocity>. The <aberration of light> maps a rest-frame ray at $\theta'=\pi/2$ to $\cos\theta=\beta$, so the characteristic cone half-angle is $\theta\simeq\gamma^{-1}$. Thus its conventional full width is $2/\gamma=2\times10^{-6}\ \mathrm{rad}$.
\b[The isotropic-emission assumption has no beam nulls.] For example, its received bolometric angular <power> is proportional to $D^4$, where the <Doppler factor> is $D=[\gamma(1-\beta\cos\theta)]^{-1}>0$. In particular it is nonzero at $\theta=1/\gamma$, disproving a literal null there. A true dipole pattern for an accelerated <electron> is not isotropic in its rest frame and needs a specified polarization and viewing plane. The width just quoted is a characteristic beaming angle, not a null-to-null width under the stated premise.
For a circular orbit seen in its orbital plane, let the <velocity> direction sweep past the observer through $-1/\gamma<\psi<1/\gamma$. The emission interval is $\Delta t_{\rm em}\simeq2/(\gamma\omega_g)=1.14\times10^{-2}\ \mathrm s$. The observed interval must also include light-travel compression:
$$
\Delta t_{\rm obs}=\frac{1}{\omega_g}\int_{-1/\gamma}^{1/\gamma}(1-\beta\cos\psi)d\psi
=\frac{2}{\omega_g}\left[\frac{1}{\gamma}-\beta\sin\frac{1}{\gamma}\right]
\simeq\boxed{\frac{4}{3\omega_g\gamma^3}=7.6\times10^{-15}\ \mathrm s}.
$$
Its ratio to the <gyromotion> period is $\Delta t_{\rm obs}/T_g\simeq2/(3\pi\gamma^3)=2.1\times10^{-19}$. Since the boundary of a smooth beam is conventional, only the scale $\omega_g^{-1}\gamma^{-3}$ is invariant under reasonable pulse-width definitions. For helical motion the viewing geometry and pitch angle are additional inputs; the numerical estimate assumes transverse motion and a sightline intercepted by the cone.<Essential Radio Astronomy by James J. Condon and Scott M. Ransom>[Chapter 5, section 5.1.3: relativistic beaming and the difference between emission duration and pulse arrival duration]{ref}
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