Is this fat pocket or muscle near mouth? Is lifting the only option? (Photo)

Impossible to give you any answer in your case because you NEED ONLY IN PERSON examinations, measurements, discussions. Again why have you NOT sought these consults? 

 Hi, I have performed many facial shaping procedures using dermal fillers, facial implants (cheek, chin), liposuction and/or facelifts for over 30 years.  From the photos, your chin is weak.  
 I have performed many Chin Augmentations using chin implants for over 30 years.  From the photos, your chin is weak.  A weak chin creates an imbalance making the nose appear larger, the mid face top heavy and the lower face look short that de-emphasizes the lips and allows early formation of a double chin and "pseudo-jowls".   Pseudo jowls differ from "jowls" in what causes them, what they represent and how they are best treated.  Pseudo jowls are defined by a fold of tissue laxity that goes from the corners of the mouth to approximately 1/3 or 1/2 the distance to the lower jaw line.  It’s caused by a weak chin and its lack of tissue support around the chin and mouth area.  It's best treated by correcting of the weak chin with a silastic chin implant.  Jowls are described below in greater detail.  Chin augmentation using a chin implant will add projection to the chin creating harmony and balance to the lower face. 
 I have found that placement of a silastic chin implant, through a small curved incision under the chin (also allows excess skin removal) to be very safe, quick, highly effective and far less invasive than a sliding genioplasty.  I perform chin implant surgery in 30 minutes or less, often using a local anesthetic alone.  In my opinion, you are a good candidate for chin implant surgery however since you had a previous chin implant removed perhaps you should consider filling in the "pseudo jowls" with an off the shelf dermal filler. 
 The back portion of your jaw line is a bit weak and can be augmented, without making your face appear masculine, using precise placement of a dermal filler (I prefer using Restylane Lyft).   In my experience and despite its recent increase in popularity, fat transfer (fat injection) offers "far" less of a reliable and predictable volume for facial shaping than an off the shelf dermal filler or facial implant.  For that reason, I do not use fat to shape the cheeks, chin, lips or jaw line.  Tissue physiology is quite simple.  Tissue requires a blood supply in and out as well as lymphatic connections to remain viable and alive.  Once fat is removed from the body all of these things have been disrupted.  Just because the removed fat is mixed with PRP or something else doesn't make the blood and lymphatics magically re-appear.  The fat at that point is not living tissue which means that it's prone to being dissolved by the body (most likely in an uneven and unpredictable manner).  Injecting fat back into the face does not create the required elements to make the fat living tissue once again.  So the argument that fat is alive and viable in the face once it's been removed and re-injected makes no sense to me as a physician and surgeon.
 The other issue that I have with fat transfer is the lack of precision.  Fat is thick by nature which means it's not the same consistency as an off the shelf dermal filler.  Fat injections use an increased volume injected in an attempt to compensate for the volume loss that "will" happen.  This means a lack of specific shape and volume that simply can not begin to compare with the specificity of using a silastic facial implant of a "known" shape and volume.  In that regard fat offers too much of an unknown to make it a reliable and predictable method for facial shaping.  There's a significant difference between a 3mm and 5mm thick cheek implant.  You can imagine the magnitude of difference there is between retaining 60% of 25cc's of fat versus 35%.  In my humble opinion, I just don’t see how fat could possibly be used to precisely shape facial features?
 If you have "jowls” (which you do not seem to have according to the photos) are sagging facial tissues and an indication for some form of a SMAS facelift.  The underlying SMAS layer, of the face, must be dissected, lifted, trimmed and re-sutured (not merely folded or suspended with threads or sutures that will not last).  The excess skin is then removed and the facelift incisions closed.
 My most popular SMAS facelift is the minimally invasive, short incision SMAS facelift that has all the benefits of more invasive facelifts (traditional, mid-face, deep plane, cheek lift and subperiosteal facelifts) but with these added benefits:
very small incisions and no incisions extend or are placed within the hair.
minimal tissue dissection = less bruising and swelling = rapid recovery ( several days instead of weeks or months with the more invasive type facelifts mentioned)
can be performed in 90 minutes or less, with or without general anesthesia
no incisions within the hair = no hair loss
excess fat can be removed from the face and neck
excess skin removed from the face and neck
cheeks, chin and jaw line can be augmented with dermal fillers (I prefer Restylane Lyft) or facial implants
most patients fly back home to parts all over the world in as little as 3 days post-op
I combine facial shaping with every facelift procedure.  When jowls are present, these should be done in concert and not alone or separately in order to create a naturally, more attractive face.
Hope this helps.