What is the rational course of action in such a situation?

Being able to cast off self-contradictions (A is equal to negation-of-A) is as close as I can offer to a knowable value that won't dissolve. But I may be wrong, depending on what you mean by sufficient deconstruction. If the deconstruction is sufficient, it is sufficient, and therefore sufficient, and you've answered your own question: we cannot know. Which leads to the self-contradiction that we know one thing and that is we cannot know any thing including that we cannot know anything.

Self-contradictions (and infinite regressions) suggest a problem with explanations, not what is explained. The rational course of action is to not confuse the world with a map of the world (in this case, rationality) even if the world does contain that map. The map is entirely inside the world, the world is not entirely inside the map.

More specifically, I spent a little time today answering this question and several hours interpreting a classroom for a deaf student. If you reverse those priorities, you get caught up in maps and don't do much to enjoy or improve this nice territory.