The best argument to be made for this idea is to point to, by way of analogy, the laceless rochet that is traditionally used by prelates in penitential seasons and in times of official mourning. In fact, this might also be part of the root of the confusion. It's an interesting analogy to be certain; the problem, however, is that while this certainly was the rule where choir vesture was concerned, it is not the case (so far as I have ever found) that it was for liturgical vestments. In fact, one need only look at images from Rome of yesteryear to see that lace was indeed used during those liturgical times and seasons.
As best I can understand it, this notion seems to rest upon the idea that lace is decorative and that is why it is felt to be out of alignment with such times of the liturgical year. Of course it is worth remembering that lace wasn't the only way these were historically given decoration; other forms included apparels, galloons or embroidery -- none of which have I ever seen included in this supposed prohibition even though they too are decorative as part of their function; an inconsistency to be sure. For that matter, the fabrics used for the primary vestments such as the chasuble were also frequently ornamental and decorative, whether by the use of brocades or the inclusion of embroideries -- or both. Then there is the liturgical metalwork. One can see how quickly this all begins to unravel -- not least of which because of the impracticality of trying to apply this (which may well be why the Church didn't).
Ultimately, if one wishes to adopt this practice of not utilizing lace for penitential seasons there's certainly nothing wrong with doing so, and you certainly have a good parallel to draw upon in the mourning dress of prelates. However in so doing one must just remember that it is purely a personal choice, not a formal liturgical requirement -- any more than preferring one shade of purple over another, preferring a simpler purple silk to a more ornamental silk brocade, or choosing to use silver instead of gold galloons for your black or violet vestments.