Without knowing who you are talking about or anything about their experience or qualifications, I couldn't really say.
I will say that when many Black Belt instructors (in other styles than BJJ) start up grappling and BJJ classes in their schools, they often wear their Black Belts. Why? Is it ego? Is it because they are still the instructor, regardless of their skill level in BJJ? Who knows?
While I don't personally agree with misrepresenting your ability to teach a particular style, I also recognize the issues associated with strapping on a White Belt alongside your students and the image it might present to new people watching who don't understand the difference. I don't see the checkered belt thing being much of a help in alleviating the problem.
I think that as long as BB Instructors are up front and honest about their knowledge, skills and abilities on the mat, wearing their earned BB shouldn't be a problem. Their should be zero confusion however, about what their Black Belt is.....and what it is not.
I had an experience several years ago where I visited a nearby Traditional school. They had started up a Grappling curriculum where they clearly did not have one before. I introduced myself to the senior student (who was a bit of a loudmouth) while the school owner taught class. I mentioned the BJJ Black Belt who I trained under and mentioned the distance to the next nearest BJJ Black Belt, some 2 hours away. The senior student shook his head and pointed his thumb to the TMA instructor who was still teaching his TMA class on the mat. Thinking that I had lost him somewhere and that he had confused what I had said, I repeated "No, I mean Brazilian Jiu Jitsu". He nodded his head and again thumbed toward his instructor. "No, I mean B-R-A-Z-I-L-I-A-N Jui-Jitsu." I added a third time. He kept insisting that the instructor had been training for 25 years and had been out to California and trained with the Gracies and had been to seminars, etc. As if that somehow qualified him as a Black Belt in Brazilian Jiu Jitsu. It wasn't long after that I found the way out.
I don't know who was behind that charade. The instructor, who's BJJ rank remained a mystery, but all his students thought he was a BB? or....his student who attempted to BS me without having any idea what he was talking about?
I have met other instructors who strapped on White Belts when it was time to grapple and do BJJ and still others who, although they wore their Black Belts, were adamant about not misrepresenting their skill as anything other than a novice.