Assume that some plausible construct is pretending to be a regular stone statue.
Is there a way for a group of PCs to detect that such a statue is in fact a construct?
Without attacking or otherwise provoking it.
Bonus points if it works against any construct, from a distance, is usable from stealth and/or if it is not stopped by a golem's magic immunity.
I have not found any good way to do it.
As a mindless creature with a construct's immunities, it does not flinch no matter what the PCs do to- or near it, and it is immune to most things that affect creatures but not objects.
As a creature, it does not register under Detect Magic. Even if you were to make the case that a craftable constructs is also magic item, assume that the GM has already ruled against Detect Magic working.
Destroying the statue does not count as detecting the construct.