  In this paper we calculate the asymptotics of the second moment of the Bessel
periods associated to certain holomorphic cuspidal representations $(\pi,
\pi')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over
$\pi$). Using these, we obtain quantitative non-vanishing results for the
Rankin-Selberg central $L$-values $L(1/2, \pi \times \pi')$, which are of
degree twelve over $\mathbb{Q}$, with concomitant difficulty in applying
standard methods, especially since we are in a `conductor dropping' situation.
We use the relative trace formula, and the orbital integrals are evaluated
rather than compared with others. Besides their intrinsic interest,
non-vanishing of these critical values also lead, by known results, to deducing
certain associated Selmer groups have rank zero.
