Can a custom chin implant improve deep labiomental fold?

It is absolutely impossible, physically and geometrically, to apply a chin implant to widen the chin and not adding a significant amount of anterior growth, it will (add anterior projection and volume).I strongly recommend you seek the help of a surgeon experienced in multidimensional genioplasties, in your case with bone supply as grafted material taken from elbow or iliac bones. This technique allows, for example, an exclusively horizontal augmentation of the chin (if this is your goal).
It is true the vast majority of surgeons use the implants instead of the osseous surgery to adjust the chin, but this is a matter just of lack of experience and comfort ease; implants play no role when there is an autologous alternative which, actually, has plenty of superior advantages.
Being an option, chin implants have serious disadvantages, especially in the mid and long term, patients are rarely happy with them after 5-6 years, all are out sooner or later. A quick research just on this website sheds light on the common, frequent and serious complications and side effects chin implants entail, like dislocation, malpositioning, infection, over or under corrections, visibility, fake looks, nerve damage, chronic pain, poor aesthetic results and in general patients' dissatisfaction; not to forget chin implants are not versatile, they are unable to provide vertical increase, horizontal axis modifications and in most cases sufficient anterior projection; as obvious as it sounds but we have to bear in mind chin implants play absolutely no role when the chin needs reduction of one of the spatial axes; actually it is very common receiving patients who seek chin implant removal and replacement by an osseous genioplasty in the same procedure.
Fillers, another tempting resource, are a very dangerous option in chin reshaping, potentially destructive, unable to reshape and enhance respecting the basic principles of anatomical beauty, leading to mid and long term serious complications (granulomas, deformities, fistulae, etc) and the worst is they can't be removed in most cases.
Burring the chin is, strictly said, osseous chin surgery but should neither be performed, ever, since just flattens the chin by destroying the natural dome of the jaw, deletes the labiomental crease and due to the detachment of the mentalis tendon this technique commonly leads to a droopy double chin, a serious aesthetic complication.
A multidimensional sliding genioplasty allows to a versatile surgeon full customization for augmenting, reducing and leveling the chin in any of its spatial axes, leading to a myriad of combinations to suit every individual's chin anatomy and facial balance requirements; you can masculinize, feminize, round or make square, increase, decrease, supply bone graft, remove bone portions, slide fragments, etc. This absolutely impossible with other method than multidimensional sliding geniplasty, a technique providing a series of advantages among which worth mentioning:-lifetime procedure, no more revision or renewal procedures, unlike implants-more natural than implants, the procedure is totally undetectable-no visibility of implant edges during gestures, expression is even improved, nicer smile-improves the double chin effect at the sub-mental muscles (floor of mouth) hammock caused by lack of tension there; also improves the shape and tension of perioral muscles and tightens up the jowls, creating a beneficial side effect in form of an indirect facelift and neck lift, all patients see their face indirectly rejuvenated; implants do not offer this feature-no future displacements or neuritis of mental nerves-no droopy double chin (the mentalis tendon is not detached)-way lower rate of postoperative complications if compared with implants, like infections, displacements, neuritis, etc-no need for complex plates and screws or teeth braces that may need future removal; just steel wires synthesis may suffice, they stay buried into bone callus producing no disturbance
The cons for the patient are essentially the costs associated with osseous genioplasties, however it sounds cheap if you consider the lifetime benefits; obviously the main disadvantage happens on surgeon's side, since it is more technically demanding than chin implants, few surgeons are experienced or comfortable with multidimensional osseous genioplasties (though I personally perform 100% of the chin procedures by this osseous method); patients have difficulty in finding surgeons experienced and comfortable in osseous genioplasties able to freely modify the chin shape and size based on their aesthetic and gender goals.
You'll find in this website hundreds of negative experiences with chin implants in the short and mid term, do read my profile's Q&A and you'll find them too. I am doing osseous genioplasties for more than 15 years with the highest success and no one patient sought afterwards switching to chin implants, totally the opposite, I did many chin implant removals and one-staged osseous genioplasties to replace the implant and enhance the result.
See the link below to find few cases of my own practice very similar to yours which I had the opportunity to operate successfully on, sharing a lot of common features with the technical problem you have posted.
If you wish better grounded opinion well lit, focused and standard images have to be assessed: frontal, both lateral and both oblique views, also from underneath.
Feel free to request any additional information from me.

A custom implant like a prejowl implant can improve the marionettes and labiomental folds. A consultation with a facial plastic surgeon will help you determine whether you need projection of the chin at the same time. Alternatively, a genioplasty operation is helpful to address projection or assymetry of a chin. 

No form of chin augmentation can satisfactorily improve a deep labiomental fold. That is because the fold represents the mentalis muscle attachment to the bone and no matter how far up you place implant material you are never going to push it out. It is an anatomic restriction. The deep labiomental fold has to be treated by a soft tissue approach not a hard tissue one.

The only way to enhance labiomental fold is to do a sliding genioplasty by downgrafting. Implant could make this fold deeper. 

We always consider the labiomental fold when performing a genioplasty. The overall image of a good chin contour extends from the lip to the base of the chin; so one must address it. A chin implant is limited in accomplishing a natural fold. Genioplasty does succeed but one often must place a limited amount of implant material 'on the ledge' of the sliding genioplasty; or simply 'on the bone under the fold'; in order to soften the curve of the fold. Filler is too short term to consider.

 Hi, when augmenting the chin, the mental crease or fold will naturally increase.  If you want this to be more shallow, simply have some dermal filler placed within the fold.  Modification of tried and true silastic chin implant styles, to affect this region, may prove problematic.   I have performed many facial shaping procedures, including Chin Augmentation with dermal fillers or silastic chin implants, for over 30 years.  When the chin is weak, this creates an imbalance making the nose appear larger, the mid face top heavy, the lower face looks short, de-emphasizes the lips and allows early formation of a "double chin".  
 Proper placement of a silastic chin implant adds forward projection to the chin thereby creating harmony and balance to the lower face.  Using the same incision, liposuction can be performed to reduce the fat and further shape the neck.  Excess skin, from below the chin, can also be removed through the same incision.  
 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 (requires extensive tissue dissection, bone cuts and placement of metal screws and plates to secure the cut segments of bone).  I perform chin implant surgery in 30 minutes or less, often using a local anesthetic alone.  
 When consulting with surgeons perhaps ask not only is your chin weak but do they have experience performing chin augmentation?  The attached video shows how I determine whether the chin is weak.  The second part of what that surgeon said is absolutely correct.  The selection of the proper size, shape and thickness as well as the proper placement of the silastic chin implant are all "key" elements in creating a naturally looking chin.  If too large of an implant is used in a woman, the chin can look relatively masculine.  That's where experience, performing chin implant surgery, comes into play.Hope this helps.