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We can overcome the previous problems by using spacetime analysis, introduced by Curless and Levoy. Instead of trying to locate the center of the beam in a single image, we try to determine for each pixel when or at which step the beam was centered there. With small steps of moving the beam, the pixel intensity as a function of time should be almost Gaussian. Now we know for each pixel when the beam was centered on it, and we can just invert that function to find out where the beam was centered at a given time step, and proceed as previously.
Here I would like to acknowledge the efforts of Pam Neal and especially of Habib Abi-Rached in the data acquisition part of my work.