Would I benefit from a fat transfer under the eyes?

Close up pictures of your eyes would be helpful. Having said that it does appear you have depressions at the lower eyelid margins which can be augmented with fillers such as Restylane or injections of fat. There are advantages and disadvantages of each. Consult with a surgeon experienced with both techniques because the injection of fat can cause significant problems if not done precisely. In patients with bags this fat can be transferred to fill the hallows but it does not appear that this technique would be appropriate for you.

what doesn't come thru in the picture is whether you have a step off and a shadow or mainly skin pigmentation or both.  Fillers and fat fix indentations. Not much fixes pigmentation. You look young and pretty. The problem with using fat is that fat grows with weight gain, sometimes without weight gain. The surgeon has no control over it. So if you gain weight, get pregnant get menopausal, the fat grows, which almost never looks good and is VERY hard to remove in that location. It can happen years later. If you must do a filler there stick with ha fillers like restylane, not juvederm. Not fat.