Is it possible to apply Lasso with weights only in penalty term or can I change the loss function in CVX? How ?
You can make the loss function whatever you want, provided it satisfies CVX’s DCP rules. There iis no reason you can;t apply weights to just the portion of the objective (loss) function you want. As to whether such a model is still considered to be LASSO, that is a statistical taxonomy question, not a CVX question.
The question of what constitutes a good loss function for your purposes, is a model formulation question outside the scope of this forum.