Though not an attorney, I spent 29+ years in resource enforcement. In my reading of this law, it would be a valid law and upheld in court. The courts would balance the definitions, goals, and limits within the law, against the individuals right of free speech. Courts have placed limits on free speech such as the classic yelling fire in a crowded location and government "top secrets".
This law was passed in 1988. It requires specific actions to be taken by the government to designate significant caves. Specific findings have to be made that certain resources are found in the cave which could be harmed. If these these are all found, the cave can be listed as significant. From what I could find the prosecution of these cases are handle via administrative civil penalities.
I would guess that for a person to claim a free speech right here, they would have to show the cave was improperly determined to be significant; the act itself was overbroad or unnecessary in its stated goals; or that they would suffer some harm by this limit on their free speech. Since this was passed in 1988, and I found ongoing code of regulations updates, an individual would have a hard time defending themselve on free speech grounds.
The big problem: how to determine if a cave has been designated significant. It appears both the Departments of Interior and Agiculture don't maintain master lists. In the case of Agriculture, I find a regulation which says each National Forest is to maintain its own list. It appears a person would have to contact the Federal agency responsible for the land the cave is on and ask.