How familiar are you with the nature of NURBS surfaces?
Here is a link to the Autodesk Alias Theory Builders. While Fusion 360 surfacing toolset is different and does not provide the level of control over surfaces Alias does, the underlying concepts and math are the same.
When done reading that you'll understand that the shape of a NURBS surface is controlled by a quadrilateral mesh of control points. In Fusion 360 and most other mainstream CAD software e.g. Autodesk Inventor, Solid Works(or so I believe) etc. usually the user does not have control over individual NURBS surface control points and surfaces are created exclusively by lofting, sweeping splines.
The geometry provided to you by the customer was not created by lofting, but most likely by manipulating individual NURBS control points, similar, for example to how this can be done with T-Splines in Fusion 360 or with a Sub-D mesh.
The "grid" you see is the original untrimmed NURBS control mesh (the screenshot is from Autodesk Alias). If this would have been created by commonly available lofting methods this would look very different. I assume that this originally was created in Rhino.

Alas, the user who created this did so clumsily and made some mistakes resulting in the very obvious crease toward the neck of the Guitar.
So to answer your initial question whether there are tools in Fusion 360, which allow you to manipulate an imported surface without having to re-loft or re-create curves. NO those do not exist.
Before you run off and try manipulating that surface in Rhino (I don't use Rhino but assume that this is possible in Rhino), I would not recommend creating this shape this way at all.
This next image shows the curvature comb of the spine in Fusion 360. The wavy nature is a result of manipulating individual NURBS control points without paying attention to curvature.

The curvature of the perimeter is very bad and is the result of trimming - likely with a spline that also has wavy curvature - across an already not so great multi-span surface. (See the Alias Theory Builders for the explanation what a span is)

This next screenshot shows the curvature of the spine sketch you created with a control point spline. You might be surprised to see that this curvature comb has a very distinct sharp point in the curvature comb.

Control point splines allow you to control curvature better than fit point splines, but only if used properly. The predominant use of control point splines in surfacing is to create single-span curves.
A 3-degree spline should contain 4 points (again, read through these Alias theory builders). You created this spline curve using a 3-degree spline with 6 points, resulting in a multi-span curve. Multi-span curves are not generally bad, but it is easy to create these breaks in curvature when controlling individual CV-spline control points. It would take a bit of finagling to smoothen this out.
In this case, you might have been better off using a 5-degree curve. With 6 points it would result in a single-span curve with smooth curvature.
Just in case you are wondering how you would create a 4-degree curve. Simply use a 5-degree curve with 5 points. The degree will then drop to 4.
Also, the fit point splines in Fusion 360 are 5-degree multi-span splines. They can create very nice curves and surfaces if you keep the number of fit points to a minimum and use the tangent handles to control the shape. And, as always, check your curvature.
In the end, there isn't really much you can do with this in Fusion 360 other than trying to re-build this from scratch with good curves and that isn't trivial. Creating good surfaces for such deceptively simple-looking surfaces is never really trivial 😉
My suggestion would be to make your customer aware of this surface flaw and manage his expectations.