Can you apply Construct only templates to a Tsukumogami, and can you make an Undead Tsukumogami?

[Reksew_Trebla]

Tsukumogami

So normally I'd say yes and yes. You just apply the Construct only templates first, and apply the Tsukumogami template before Undead templates.

But the problem I'm having is the lore, which should affect RAI. While Tsukumogami is applied to Animated Objects as an acquired template, this is only because regular objects don't have a stat block as creatures. The actual lore is that Tsukumogami used to be regular objects, not Animated Objects.

Since they technically were never a Construct to begin with, only being one for the purposes of applying the template, this would suggest you could not apply Construct only templates.

For a similar reason, I am unsure if they could become undead. While you could say they are now Outsiders (kami, native), the living part is strictly the kami part, as without it, they are just regular, impossible to be undead, objects.

I know RAW this template stacking works, it's just the RAI for the lore that I am unsure about.

[Diego Rossi]

The Tsukumogami loses its previous Type and receives the "outsider (kami, native)" type. That should invalidate the templates that require it to be a construct.

For the Undead part, it is even more dicey. When the Tsukumogami dies it leaves behind an object, probably a broken object, not a corpse. As they are native outsiders, they have a soul.
As I see it, they can't become corporeal undead (there isn't a body to animate), but they can become incorporeal ones.
AFAIK, things like the "ghost" of a writing brush exist in Japanese folklore (more as a spirit), so a spirit Tsukumogami seems appropriate.