= Solution
Power is energy per time. In <geometrized units>, energy has the dimension of length and time is converted to length using $c$, so the dimensionless <geometrized luminosity> is
$$
\boxed{L_{\odot,\rm geom}=\frac{G L_\odot}{c^5}\simeq1.06\times10^{-26}.}
$$
Equivalently, $c^5/G$ is the corresponding SI power unit. The lifetime for emitting the entire <rest energy> at the given constant <luminosity> is
$$
t_{\rm all}=\frac{M_\odot c^2}{L_\odot}\simeq4.65\times10^{20}\,\mathrm s
=\boxed{1.47\times10^{13}\,\mathrm{yr}}.
$$
The same result follows from $M_\odot^{(\rm time)}/L_{\odot,\rm geom}$. \b[The stipulated full-mass radiation time is of order $10^{13}$ years]; this is the constant-luminosity energy budget asked for, not a stellar-evolution calculation.
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