Why Would Pre Filtering Measurement Data Affect the Least Squares Estimate?
I'm not sure what's you model is.

Let's say it is something like:

$$ y = H x + n $$

Now, using the Least Squares model is optimal (In the MSE sense) when $ n $ is AWGN (It is the linear optimal estimator if the noise is white).

So unless the noise in your model is colored, no gain by filtering the data before applying the Least Squares method.

Now, what if the noise isn't IID White Noise?


If it is white just with different statistics, then use the Weighted Least Squares.
If it is colored you should apply Whitening filter on the data.


So, if the Whitening filter is given by $ W $ then you have:

$$ z = W y = W H x + W n = A x + v $$

Now you have classic LS model, where your model matrix is $ A = W H $ and your white noise is $ v = W n $.
Now you can just apply the regular Least Squares.