Would a chin surgery help my nose look smaller/balance my face? (Photo)

You need forward projection to reshape your chin profile but also vertical reduction, a very common combination in the office of the genioplasty surgeon: anteriorly short + vertically long chin bone structure.In the overlapping genioplasty you perform a complete horizontal cut just below the mentalis nerves point of exit and through the whole chin and its side wings, the mobilize the cut fragment attached to the oral muscles, and finally overlap the fragment so that the back cortex of the chin jumps onto / makes contact with the anterior surface of the jaw; by means of this witty maneuver the chin is forwarded and at the same time vertically shortened, which is what you need.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.

The very short answer to your question is Yes!!!From the front, the lower third of your face is elongated because one's eye travels to the very bottom of your retrusive mandible, i.e. chin.  A chin implant and/or sliding genioplasty will not only balance your face, shorten the perceived distance to the bottom of the lower third of your face, give support to your lower lip so that you overcome your lip incompetence, and give a very strong and virile appearance both in front and in profile view.I recommend you consult with two or three Board Certified plastic surgeons demonstrating excellent before and after pictures of chin augmentations on their websites.  

Based on your photo I think you would be a good candidate for a chin implant to balance your face.  Having a stronger chin tends to balance out a strong nose.

It is common to have facial imbalance with Central prominence around the midface and nose, and weakness or lack of projection in the lower face and chin. Correcting this can be done either by reducing the prominence of the nose, increasing projection of the chin, or both. The decision between these options is based on your needs and recommendations of your surgeon. Looking at your pictures, I think that sliding genoplasty would be an excellent option for you and would significantly improve your facial balance. Following this, if you still felt that there was imbalance you could consider rhinoplasty as a second operation later.

There is no question that your shorter and vertically long chin makes your nose look bigger than what it would if the lower third of your face was in better proportion to the rest of your face. This could be visually illustrated with computer imaging. The only debate is whether the chin augmentation should be done by an implant or a sliding genioplasty and whether you want the rest of the jawline augmented at the same time as well.

 The photographs demonstrate an overly projecting nose, and an under projecting chin.  A rhinoplasty procedure can accomplish refinement of the nose and de-projecting it and making it smaller.  Placement of a chin implant will augment the chin forward for better facial balance and portions, especially with respect to the nose.  Both procedures can be performed separately or together.  For many examples of this combination, please see video below

Your perspective is quite valid. A genioplasty would help the balance of the profile but you need to consider the frontal view as well in your case since your facial growth pattern has a narrow long taper frontally. So the width of the jawline must be considered for a natural result