First, if your images normally look like that, your sensors are deeply saturated, and consequently your 3D ranges are either worthless or much less accurate than they could be.
Second, you should aim for matching one point per rectified scanline on each image of the pair, rather than a set of points. The whole idea of using a laser stripe is to get a well focused beam of light on as small a spot or band as possible, so you can probe the surface in detail.
For best accuracy, the peak-finding should be done independently on each scanline of the original (distorted and not rectified) images, so it is not affected by the interpolation used by the undistortion and stereo rectification procedures. Rather, you would use the geometrical undistortion and stereo rectification transforms to map the peaks detected in original images into the rectified ones.
There are several classical algorithms for peak-finding with laser stripe-based triangulation methods, you may find this other answer of mine useful.
Last, if your setup is expected to be as in the picture, with the laser stripe illuminating two orthogonal planes in addition to the object of interest, then you do not need to use stereo at all: you can solve for the 3D plane spanned by the laser stripe projector and triangulate by intersecting that plane with each ray back-projecting the peaks of the image of the laser stripe on the object. This is similar to one of the methods J. Y. Bouguet used in his old Ph.D. thesis on desktop photography (here is a summary by S. Seitz). One implementation using a laser striper is detailed in this patent. This method is surprisingly accurate: with it we achieved approximately 0.2mm accuracy in a cubic foot of volume using a dinky 640x480 CCD video camera back in 1999. Patent has expired, so you are free to enjoy it.