Let me make some very general observations that address some of your questions, including the switching possibilities you mention at the end.

First, I claim that under the hypothesis of projective absoluteness, we have your desired absoluteness for countable structures. Projective absoluteness, a consequence of the existence of sufficient large cardinals, asserts that the truth value of projective statements (with real parameters) cannot change by forcing. But the point now is that for countable $A$ and $B$, the assertion $A\equiv_{II} B$ is determined by projective statements, since each second order assertion is essentially quantifying over the subsets of $A$ or $B$ and hence by countability equivalent to quantifying over the reals. So if we cannot change projective statements with parameters coding $A$ and $B$, then we cannot affect the truth of any second-order assertion in $A$ or $B$, and so $A\equiv_{II} B$ is absolute to forcing extensions.

Second, and a bit more generally, I claim that if the maximality principle $\text{MP}(\mathbb{R})$ holds and the truth value of $A\equiv_{II} B$ can change at all by forcing, where $A$ and $B$ are countable, then in fact we get the full switching behavior that you asked about; we can keep switching on and off as long as desired (an in particular, attain the even/odd pattern at the end). The maximality principle (see my paper A simple maximality principle, also independent work of Stavi and Väänänen) asserts that any forceably necessary statement (with real parameters) is already necessary, or in other words, if $\varphi(z)$ is a statement which can be forced in such a way that it remains true in all further forcing extensions, is already true in all forcing extensions. In this case, the point is that if $A$ and $B$ are countable, then we can use them as parameters, and if $A\equiv_{II} B$ cannot perpetually be changed by forcing, then either $A\equiv_{II} B$ or $A\not\equiv_{II} B$ is a forceably necessary statement, since we can make it stabilize, and so by $\text{MP}(\mathbb{R})$ the statement must therefore already be true and unchangeable by forcing.

Let me make some very general observations that address some of your questions, including the switching possibilities you mention at the end.

First, I claim that under the hypothesis of projective absoluteness, we have your desired absoluteness of second-order elementary equivalence for countable structures, to all forcing extensions. Projective absoluteness, a consequence of the existence of sufficient large cardinals, asserts that the truth value of projective statements (with real parameters) cannot change by forcing. But the point now is that for countable $A$ and $B$, the assertion $A\equiv_{II} B$ is determined by projective statements, since each second order assertion is essentially quantifying over the subsets of $A$ or $B$ and hence by countability equivalent to quantifying over the reals. So if we cannot change projective statements with parameters coding $A$ and $B$, then we cannot affect the truth of any second-order assertion in $A$ or $B$, and so $A\equiv_{II} B$ is absolute to forcing extensions.

Second, and a bit more generally, I claim that if the maximality principle $\text{MP}(\mathbb{R})$ holds and the truth value of $A\equiv_{II} B$ can change at all by forcing, where $A$ and $B$ are countable, then in fact we get the full switching behavior that you asked about; we can keep switching on and off as long as desired (an in particular, attain the even/odd pattern at the end). The maximality principle (see my paper A simple maximality principle, also independent work of Stavi and Väänänen) asserts that any forceably necessary statement (with real parameters) is already necessary, or in other words, if $\varphi(z)$ is a statement which can be forced in such a way that it remains true in all further forcing extensions, is already true in all forcing extensions. In this case, the point is that if $A$ and $B$ are countable, then we can use them as parameters, and if $A\equiv_{II} B$ cannot perpetually be changed by forcing, then either $A\equiv_{II} B$ or $A\not\equiv_{II} B$ is a forceably necessary statement, since we can make it stabilize, and so by $\text{MP}(\mathbb{R})$ the statement must therefore already be true and unchangeable by forcing.