I don't think interpolating your data will fix the problem, as your data is seriously bumpy.
Interpolating that mess will not fix it. You cannot make noisy data smoother by interpolating it. It will still have the same bumps in it, though it will look less jagged.
You need to smooth it, but you want to maintain that one contour. Sigh. Smoothing in any form is a process that will see any variation as noise, and then produce a surface with less variation. But any transitions will be interpreted as potentially noise too. So any smoothing you do will tend to kill off the shape of that contour.
Could a smoothing tool be written that would maintain a given contour? Well, yes, I know that to be true. For example, my gridfit utility (found on the File Exchange) can smooth a surface like your data. And I did once show such a tool (in 3-d) can be modified to follow a general prescribed path. The code itself gets a bit nasty, since you would need to define that path, then build the linear system to incorporate that path as a constraint. Possible, but it takes some work. And the path constraint introduces other artifacts into the fit itself, which makes the result less than a happy one.
A big problem is, that contour you show includes a very narrow (and deep) valley. And as soon as you start to smooth the surface, that valley starts to disappear.
Here, I've overlayed a mesh (in red) of a smoothed version of the surface on top of the old one.
But there is no simple way to control what happens along that curved path you have indicated. I don't know what contour level you chose, else I could show how the shape of that same contour has changed. Anyway, any contour plot of your original data will be quite bumpy, so the curve you have drawn surely does not come from a contour taken from that noisy data.
