= Solution
For a trajectory connecting the degenerate minima in infinite Euclidean time, the first integral gives $m\dot q^2/2=V(q)$. The <quartic double-well instanton> and its reverse are
$$
q_I(t)=a\tanh\frac{\omega(t-t_0)}2,\qquad q_{\bar I}(t)=-q_I(t).
$$
Their action is
$$
\boxed{S_0=\int_{-a}^a\sqrt{2mV(q)}\,dq=\frac{m\omega}{2a}\int_{-a}^a(a^2-q^2)\,dq=\frac23m\omega a^2.}
$$
The <instanton fluctuation prefactor> $K$ has dimensions of inverse time. It incorporates nonzero Gaussian fluctuation <eigenvalues> and the translation zero-mode Jacobian; a conventional <determinant> expression is
$$
K=\sqrt{\frac{S_0}{2\pi\hbar}}\left[\frac{\det(-\partial_t^2+\omega^2)}{\det'(-\partial_t^2+V''(q_I)/m)}\right]^{1/2},
$$
with common endpoint regularization and the prime removing the translation mode. Only its positive <quantum tunnelling> rate $\kappa=Ke^{-S_0/\hbar}$ is needed here.
In the <dilute instanton gas>, well-separated crossings alternate in direction. Integrating $n$ ordered centers gives $\tau^n/n!$. A path returning to the same well has $n$ an <even number>, and one connecting opposite wells has $n$ an <odd number>. The common leading harmonic endpoint factor follows from the supplied <harmonic oscillator transition kernel>:
$$
\mathcal A_0=\sqrt{\frac{m\omega}{\pi\hbar}},\qquad
\begin{aligned}
\langle a|e^{-H\tau/\hbar}|a\rangle&\simeq\mathcal A_0e^{-\omega\tau/2}\cosh(\kappa\tau),\\
\langle a|e^{-H\tau/\hbar}|-a\rangle&\simeq\mathcal A_0e^{-\omega\tau/2}\sinh(\kappa\tau).
\end{aligned}
$$
Expanding these functions gives the sums over <even numbers> and <odd numbers>, with the position-kernel normalization $\mathcal A_0$ that its abbreviated formula suppresses. Validity requires $S_0/\hbar\gg1$, $\omega\tau\gg1$ and $\kappa\ll\omega$, so typical crossing separation greatly exceeds an <instanton> width. Local anharmonic corrections replace the harmonic well energy by its perturbatively corrected value; the leading formula does not claim uniformly negligible relative error at arbitrarily large $\tau$.
The two exponents identify the <double-well tunneling splitting>:
$$
\boxed{E_{\rm even}\simeq\hbar\omega/2-\hbar\kappa,\qquad
E_{\rm odd}\simeq\hbar\omega/2+\hbar\kappa,\qquad\Delta E=2\hbar\kappa.}
$$
The superposition with an <even> spatial <wavefunction> is the lower state. Coherent <quantum tunnelling> removes the classical degeneracy, and the exponentially small splitting sets the long <quantum tunnelling> timescale.
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