Yes.
So the issue is that the object does not have the shapechanger tag, which is a specific tag associated with certain creatures such as vampires and were-things.
This means that yes, you can polymorph the object into a shapechanger. However, immediately after doing so, the creature has the shapechanger feature and it would automatically succeed on the saving throw against being polymorphed any longer. Or it would, except there's a caveat in the spell itself that prevents that.
From the True Polymorph spell, emphasis mine:
Shapechangers aren't affected by this spell. An unwilling creature can make a Wisdom saving throw, and if it succeeds, it isn't affected by this spell.
The creature is friendly to you and your companions. It acts on each of your turns. You decide what action it takes and how it moves.
So, since the object isn't a creature to begin with, there's no saving throw. Then, when it becomes a creature, it's friendly to you and you decide what action it takes. This means it is not an unwilling creature unless you say so, negating the need for the save at all. Any ruling that says the spell needs to be "re-applied" after the polymorph is strictly house ruling.
So by a strict RAW reading, the saving throw only happens on the initial casting of the spell. So for as long as the spell is maintained on the object, it does not get to automatically pass the save because an object doesn't get to save on the initial cast.
Frankly, at the end of the day this isn't OP and I would permit it at my table. There are plenty of far more powerful uses for this spell, including using True Polymorph to permanently change into an Ancient Brass Dragon (Level 20 character becomes a CR 20 dragon). They do not have the shapechange tag, yet they do have change shape. This specific ability permits them to become any CR 20 or lower humanoid or beast as an action as many times as they want.