Question

Hello and greetings!
It is well known that there is a dispute between Ba'al HaMaor and the Milhamot as to whether the victim of theft has the power to consecrate the object when the thief is not refusing. Avnei Milu'im (28:13) argues that even Nachmanides agrees that the object can be transferred to the thief himself, because for the thief the object is considered something in his possession.a0
Seemingly, Avnei Milu'im's words prove that "not in his possession" is not a rule specifically about consecration or ownership, but rather about the laws of acquisition. It should be added that it sounds from his words as though the deficiency is not on the side of the transferor, but on the side of the recipient, who cannot acquire an object that in the end does not come to him; and therefore, when the object is already in the possession of the recipient, there is no problem.
If I understood Avnei Milu'im correctly, it would seem one can challenge his view, since the main rule of "not in his possession" was stated regarding consecration, and yet with consecration too, wherever it is, it is in the treasury of the Merciful One, and the object is in His possession. If so, why can the object not be consecrated?