I'm not arguing with you that he actually said that, but we did wear the white helmets under Perkins at least once when playing teams that did not have dark colored helmets. I remember attending the 1983 Alabama v. LSU game in Tiger Stadium, and we wore white helmets then. LSU, of course, was wearing their usual gold helmets, which is far from a dark color. That game was on ABC, if I remember correctly, and you could probably get the film from Simon Bott.
The teams we often wore it against -- MSU and Vandy come to mind -- generally had darker colored helmets (Vandy had a black helmet at the time), but that wasn't always the case.
The white helmet actually has a pretty interesting history. Most people do not remember this, but at times in the early 1960's Bryant actually used the white helmet for kicking specialists while the rest of the team wore the usual crimson helmets. And, another forgotten fact, we actually wore white helmets in 1969 with the 100th anniversary of college football design.