  Vibrational branching ratios in the B$^2\Sigma^+$ -- X$^2\Sigma^+$ and
A$^2\Pi$ -- X$^2\Sigma^+$ optical-cycling transitions of BaH molecules are
investigated using measurements and {\it ab initio} calculations. The
experimental values are determined using fluorescence and absorption detection.
The observed branching ratios have a very sensitive dependence on the
difference in the equilibrium bond length between the excited and ground state,
$\Delta r_e$: a 1 pm (.5\%) displacement can have a 25\% effect on the
branching ratios but only a 1\% effect on the lifetime. The measurements are
combined with theoretical calculations to reveal a preference for a particular
set of published spectroscopic values for the B$^2\Sigma^+$ state ($\Delta
r_e^{B-X}$ = +5.733 pm), while a larger bond-length difference ($\Delta
r_e^{B-X} = 6.3-6.7$ pm) would match the branching-ratio data even better. By
contrast, the observed branching ratio for the A$^2\Pi_{3/2}$ -- X$^2\Sigma^+$
transition is in excellent agreement with both the {\it ab initio} result and
the spectroscopically measured bond lengths. This shows that care must be taken
when estimating branching ratios for molecular laser cooling candidates, as
small errors in bond-length measurements can have outsize effects on the
suitability for laser cooling. Additionally, our calculations agree more
closely with experimental values of the B$^2\Sigma^+$ state lifetime and
spin-rotation constant, and revise the predicted lifetime of the H$^2\Delta$
state to 9.5 $\mu$s.
