Short sketch #5: Possession and ergative syntax (again)

I have recently described something along the lines of a split-dechticaetiative system, and split-ergative should be familiar to pretty much every conlanger by now, I guess. How about split-possessive systems?

Some languages, such as English, mark possession on the owner, others mark it on the possessum, some mark it on both, some don't mark either; some have other systems than those even. How about a system that utilizes different approaches in different circumstances.

Let's say the language normally marks the possessum, but in transitive subjects marks the possessor.

John house-[poss] is big

We burned down John house-[poss] 

vs.

John-[gen] dog attacked us!

Let us imagine some further complications:

1.  Transitive subjects possessing the objects: John built house-[poss] by himself
2. Possessors of transitive subjects possessing the object as well: John-[gen] dog attacked cattle-[poss]

Alternatively we could turn it all around:

John house-[poss] is big ← with free word order this'd be cool

We burned down John-[gen] house

John dog-[poss] attacked us!

and turn the complications around as well, although getting a slightly weird thing:

1. John-[gen] built house
2. John dog-[poss] attacked cattle-[poss]? Joh-[gen] dog-[poss] attacked cattle?

The last option seems unnatural somehow, in a way the others don't quite.

Further, let's have -[poss] be identical to the instrumental marking in the language.