Can Massaging Reverse Fat Injection Effects on my Lips?

Massage will not change the amount of fat in the lips after fat injection. The most likely reason for the swelling so early is edema and this will resolve with time. The lips will gradually improve with time in terms of size. Massage, however, will minimize the risk of lumps and uneveness.
I always use BruisePak tablets after fat injections for this reason.

It will certainly take longer than 12 days for the fat swelling to resolve. Furthermore, unless the technique was the Viafill system, a large amount of the fat will resolve and go away. So be patient. It will get better.

Being that you are only 12 days out from fat injection to the lips, I would wait it out a bit. They are still swollen from the procedure and the swelling will come down more.

You are ONLY 12 days after the transfer. There is still a lot of swelling and healing going on.
Wait at least 12 weeks. Massage will not help, may cause more swelling, wait sometimes a short course of steroids can decrease the swelling and inflammation.

As long as the swelling is decreasing, you can consider a seven day course of oral steroid per your doctor. I find that commonly, some swelling is still present in week 3.
Oral and topical Arnica is part of my perioperative care. Gentle massage can help with swelling a bit of shaping.
Fat grafting has an important technical know-how.

Massage is not likely to make a huge difference and at best it will make the swelling go down faster. Your upper lip should never be bigger let alone greater than 1/2 the size of your lower lip. My theory on aesthetics or facial beauty, Circles of Prominence, has discovered that your upper lip should be about 1/2 the size of your lower lip. Your swelling may go down quite a bit. Hopefully most of that will go away.
Give it some time and try not to worry too much about it. You can make the fat go away through microliposuction with the same cannulas that you put them in with. Sometimes you can also excise the extra fat out and different types of excisions can be done to make the lip look better and smaller. The swelling will likely go down quite a bit though. It can be done though!