= Solution
A fixed distant observer receives appreciable radiation while the velocity points within an angle of order $1/\gamma$ of the line of sight. The corresponding emission interval is $\Delta t_{\rm em}\sim R/(\gamma v)$. The source moves toward the detector during that interval, shortening the light-travel distance. In the far-field limit,
$$
\frac{dt_{\rm arr}}{dt_{\rm em}}=1-\beta\cos\phi,
$$
where $\phi$ is the angle between velocity and the detector direction. Near the tangent, $1-\beta\cos\phi\sim\gamma^{-2}$, so
$$
\boxed{\Delta t_{\rm arr}\sim\frac{R}{c\gamma^3}
=9.25\times10^{-18}\ \mathrm s.}
$$
The emission interval itself is of order $3.2\times10^{-10}$ s. Omitting the arrival-time compression would therefore miss many orders of magnitude.
For a definite angular window $-1/\gamma\leq\phi\leq1/\gamma$, integrating the arrival-time relation gives
$$
\Delta t_{\rm arr}=\frac Rv\left[\frac2\gamma-2\beta\sin(1/\gamma)\right]
\simeq\frac{4R}{3c\gamma^3}=1.23\times10^{-17}\ \mathrm s.
$$
The order-one coefficient depends on how the beam window or pulse width is defined. These are intrinsic single-electron radiation-flash scales. For <bunch-length broadening of synchrotron radiation>, a finite bunch moving through the same tangent point contributes an arrival-time spread of order its length divided by $v$; the longitudinal bunch profile and instrumental response must be convolved with the intrinsic flash. “Short bunch” alone does not specify an experimental bunch pulse duration at the attosecond scale, because no bunch length has been supplied.<Synchrotron Radiation by Richard Fitzpatrick>[§12.22, equations (1974)–(1981), beaming and arrival-time compression]{ref}
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