= Solution
The peak near $0.5\,\mu{\rm m}$ can be reflected starlight carrying the stellar spectral shape, while the $1\,\mu{\rm m}$ peak can be thermal emission from a very hot, young, or strongly irradiated planet. <Wien displacement law> then suggests $T_s\simeq5800\,{\rm K}$ and $T_p\simeq2900\,{\rm K}$, so $T_p/T_s\simeq1/2$.
Assume both objects are in the far-infrared <Rayleigh-Jeans law> regime. From
$$
10^{-3}\simeq
\left(\frac{R_p}{R_s}\right)^2\frac{T_p}{T_s}
$$
one obtains $R_p/R_s\simeq\sqrt{2\times10^{-3}}\simeq0.045$. With $R_s=0.5R_\odot$ and $R_J\simeq0.10R_\odot$,
$$
\boxed{R_p\simeq0.22R_J},
$$
of order a Neptune radius. This estimate is sensitive to the peak interpretation and blackbody assumptions.
If the system transits, secondary-eclipse emission spectroscopy is especially effective because it separates planetary light from starlight and directly measures the favorable $10^{-3}$ infrared contrast. For a sufficiently wide orbit, direct-imaging spectroscopy is preferable because the hot planet is self-luminous and can be spatially separated from its star.
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