Pullam and Huddleston’s explanation, the "oblique genitive" is more restricted […] the genitive noun phrase and the head of the enclosing noun phrase is vague and unhelpful as it does not specify the relationship: all it does is restate the reader’s suspicions (those suspicions of the reader’s) and validate those suspicions – by “authority”. It is, in fact, the equivalent of “depends on the context”, which we knew anyway.
I suggest the context is "possession."
There is an excellent article in WORD, Volume 6, April 1950, Number 1 The English Construction A Friend Of Mine by Anna Granville Hatcher that gives the history of thought on, and use of the double genitive.
Although it does not mention things or animals, throughout, it is agreed by all that “of mine/his/theirs/yours” refers here to ownership or at least, current possession.
The question is thus whether “That X, An X, Xs, X of mine/yours/his/hers/ours/theirs/John’s” can be used with a noun or pronoun that is culturally incapable of possessing something.
*A leaf of the tree’s fell to the ground.
*A leaf of its fell to the ground.
*One of the roofs of the building’s is leaking.
*Those roofs of the building’s are leaking.
Culturally, in most of world in which English is the first language, dogs hover, anthropomorphically, on the boundary of having possessions. This anthropomorphism is illustrated in:
*I’ve given the dog another bone because it’s lost one of its.
But
"I’ve given the dog another bone because he’s lost one of his.
And
*“That tail of its is wagging.”
But
"That tail of hers is wagging.”
However, "a bone/toy of my dog's" creates a problem of possession: "my" indicates that you possess the dog and it is thus clear that the dog has not been anthropomorphised. The phrase does not therefore work.