Could symmetry possibly break down?

Certainly. Several papers have been published over the decades which talk about the exceptions. But they are generally just refinements or quibbles aimed at getting a paper in a publication, not contradictions.

For example, monopolists may not be fully affected as expected. The effects of import taxes (tariffs) and export taxes will tend to have different time lags, so the end result on terms of trade might ultimately be the same but might not hit equally quickly. Capital controls would change things, as would price fixing laws, and fixed exchange rates. A big swing in the capital account will change things, not just the symmetry theorem of course.

...Lerner Symmetry being strongest when policies are stable or even permanent vs. let’s say if, in this case, tariffs are temporary....

The theorem holds primarily when the tariffs are permanent, and (perhaps more importantly) seen to be permanent. Nobody is going to build a factory to offset the cost of tariffs if they think the tariffs won't last a really long time.

The general rule has been robust, though. In real world countries, neither export taxes nor import taxes will reduce a country's trade deficit, and they both have the same effect of reducing both the volume of exports and the volume of imports even in the absence of policy changes or "retaliation" by trading partners. Other things being equal (though they never are), both policies will drive up the country's exchange rate and lower the real GDP.

Jim