Is Stored Fat Safe to Use for Fat Transfer?

The longer fat is stored, the more of it dies, and the more temporary the results from fat injection are.
Many patients believe that when fat is taken from one part of the body and placed elsewhere, that fat will "take".
This is not true.
Most fat that is harvested from one are of the body is dead. Fat is delicate biologic tissue that exists in a complex biologic environment in the body. Fat injection technique works by first aspirating (sucking) the fat out of the body with a vacuum (syringe), purifying it in some fashion, and then injecting it back into the patient.
The percentage of live cells is debatable. In our research (which we have presented at national meetings and published in Plastic and Reconstructive Surgery journal), the viability is 20%. The rest of the fat cells are dead and are either metabolized away or walled off eventually. In other studies, the viability rates are as low as 3% if you measure the metabolism of the cells (ability to "breathe").
We prefer LiveFill grafts, nontraumatized fascial fat grafts, for volume augmentation in the face, since they are alive. The long term survival is far higher than that of traumatized grafts.
Also, the swelling at the time of insertion is less than that of fat injection because the body does not have to deal with metabolizing away a large amount of dead cells. The disadvantage of LiveFill is its greater cost than fat injection.
If you do choose to have the fat transplantation method, the more time the fat sits in the refrigerator, the more of it dies.
Another consideration:
Although unlikely, it is possible that the patient's fat samples could be mixed up with those of another patient. Such errors are possible even in the most highly monitored blood banks at the best institutions, so they are possible in an unmonitored doctor's office. For that reason, to minimize the risk to patients, we do not store patient's samples for reinjection.

Frozen Stored Fat Transfer - Probably not in your best interest
The options your doctor gave you are not standard, are experimental, and and in short are not in your best interest. Is he/she board certified by the American Board of Plastic Surgery - I doubt it. The reason is that as a rule, the longer the fat is outside your body the more of a chance there is of fat cell death, infection from contamination, and less chance of survival. The best chance for maximum fat survival is immediate fat transfer. Even then there is incomplete survival.

Remember there is Zeltiq CoolScultpting which essentially cools fat in your body externally to make fat cells die early. It therefore doesn't make sense that cooling fat will also increase the survival of fat. See a board certified plastic surgeon with experience in this technique

Fat grafting is fickle. Frozen fat would not be expected to have the same viability as freshly harvested fat.  A better strategy would be to harvest only the amount of fat that is being transferred in the same setting.