= Solution
For a linear polyatomic <molecule>, the first-order <polyatomic vibration-rotation interaction> is
$$
B_{\mathbf v}=B_e-\sum_i\alpha_i(v_i+d_i/2),
$$
where $d_i$ is the degeneracy of <normal mode> $i$. The bending mode has $d_2=2$, so the two measured states imply
$$
\alpha_2=B''-B'\simeq-0.00429\ \mathrm{cm}^{-1},\qquad
B_e=B''+\alpha_2+\frac{\alpha_1+\alpha_3}{2}.
$$
The two stretching zero-point contributions are not supplied. \b[The true all-mode equilibrium constant is therefore not uniquely determined by this band alone.] Neglecting those unknown stretching corrections gives the bend-only estimate
$$
\boxed{B_e^{(\rm bend)}\simeq2B''-B'=1.47357\ \mathrm{cm}^{-1}.}
$$
Using $B=h/(8\pi^2cI)$ and converting inverse centimetres to inverse metres,
$$
I_e^{(\rm bend)}=\frac{6.62607015\times10^{-34}}
{8\pi^2(2.99792458\times10^8)(100)(1.47357)}
\simeq\boxed{1.900\times10^{-46}\ \mathrm{kg\,m^2}.}
$$
If the intended course approximation instead treats the observed vibration as a single nondegenerate oscillator with $B_v=B_e-\alpha_e(v+1/2)$, its extrapolation gives
$$
B_e^{(1D)}=\frac{3B''-B'}2=1.47571\ \mathrm{cm}^{-1},\qquad
I_e^{(1D)}=1.897\times10^{-46}\ \mathrm{kg\,m^2}.
$$
That is a useful effective one-coordinate estimate, but it should not be identified with an exact equilibrium constant of the doubly degenerate polyatomic bend. Both estimates give \b[$I\simeq1.90\times10^{-46}\ \mathrm{kg\,m^2}$] at the accuracy warranted here.
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