I would take random samples of the 30 input points and compute the homography in each case along with the errors under the estimated homographies, a RANSAC scheme, and verify consensus between error levels and homography parameters, this can be just a verification of the global optimisation process. I know that might seem unnecessary, but it is just a sanity check for how sensitive the procedure is to the input (noise levels, location)
Also, it seems logical that fixing most of the variables gets you the least errors, as the degrees of freedom in the minimization process are less. I would try fixing different ones to establish another consensus. At least this would let you know which variables are the most sensitive to the noise levels of the input.
Hopefully, such a small section of the image would be close to the image centre as it will incur the least amount of lens distortion. Is using a different distortion model possible in your case? A more viable way is to adapt the number of distortion parameters given the position of the pattern with respect to the image centre.
Without knowing the constraints of the algorithm, I might have misunderstood the question, that’s also an option too, in such case I can roll back.
I would like to have this as a comment rather, but I do not have enough points.