= Solution
Write $K\propto m^\alpha R^\beta h^\delta$. Matching dimensions to $ML^2T^{-2}$ gives
$$
\delta=2,\qquad\alpha=-1,\qquad\beta=-2,
\qquad K=C\frac{h^2}{mR^2}.
$$
The dimensionless factor $C$ cancels when the same confinement geometry and state are scaled. Hence
$$
K_p=K_e\frac{m_e}{m_p}\left(\frac{R_e}{R_p}\right)^2,
\qquad
\boxed{K_p\simeq8.3\,\mathrm{MeV}.}
$$
Using $m_p/m_e\simeq1836$ gives this value. It remains small compared with the <proton> <rest energy>, so the nonrelativistic mass-radius scaling is self-consistent.
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