One way to get the weights out from the linear layer is simply to pass unit vectors as input together with the actual features. To make this work we need to have weight sharing of the LinearLayer weights across these multiple inputs. This is achieved by the NetMapOperator as mentioned in the documentation.

In[371]:= With[{w = {{1, 2}}},
   net = NetMapOperator[
      LinearLayer[1, "Input" -> 2, "Weights" -> w, "Biases" -> None]];
   ]

  In[372]:= net[{{2, 3}, {1, 0}, {0, 1}}]

  Out[372]= {{8.}, {1.}, {2.}}

PartLayer[1] and PartLayer[2;;3] acting downstream of above in a NetGraph will let you siphon out the relevant pieces from which you can build your error function. Let me know if you need more help.

Just FYI, LASSO and many other regularizations are now available in Fit, which were added in Wolfram Language version 12.0 as an application of convex optimization. Ridge regression is there listed as Tikhonov regularization.

http://reference.wolfram.com/language/ref/Fit.html

If it helps, I have some sense that NetArray and NetArrayLayer, introduced in 12.2, might be involved. But how? The documentation for those two experimental functions are somewhat underdeveloped.