How to Create a Convolution Matrix with a Variable Condition Number (CN)
Let's try to think of it intuitively.
Given the LPF what would make the inverse "Hard"?  

The sections we won't be able to recover are where the LPF is 0, since there is no inverse for that we can multiply the result by.
Real world LPF won't reach "Real" zero as usually.
But what would make it hard to recover is where there are big ratios between the biggest and the smallest magnitude.
As close and fast your LPF goes to zero (And has magnitudes which are big) the harder the recovery will be.

Now just build analog LPF which this requirements at different scales, digitize it, create the Convolution Matrix and there you have it...

A practical example is given in my answer to 1D Deconvolution with Gaussian Kernel (MATLAB).