I think it's a non-sequitur to arrive from the authors' statement about White registrants benefitting from acknowledging their White privilege to the conclusion the authors would blame the discrimination White registrants have experienced on them not having read enough academic literature about racism while living under a bridge.
An interpretation that would make more sense to me is that the authors suggest that for building alliances, a good first step is often to look for similarities with those you hope to join your cause. Someone needs to make the first step and say "Hey, I see you're oppressed, too". How do you get gay activists to listen to you? Probably by mentioning stuff like heteronormativity in your texts, websites and speeches. How do you get feminists to listen to you? Likely by talking about misogyny.
After all, surely one would want to avoid painting registrants living under a bridge as generally bitter people who deny or don't care that anyone else suffers? One can disagree with the authors' argument registrants would significantly benefit from talking more about the oppression of others, but I don't see how their statement of White registrants enjoying White privilege would be an attack or accusation, rather than an observation. The way many researchers conceptualize privileges is that no matter how oppressed you may be in other areas of life, as long as you belong to a privileged group, you enjoy their specific privileges to at least some extent. Consider e.g. common statements like "All male people have male privilege.". The authors seem to simply follow this academic convention. Privilege is then not something one is usually guilty of having or able to get rid off, with perhaps the most notable exception being that of being rich - a privilege that unlike most others a person can and should get rid off. Another privilege one can get rid off is being able-bodied, though of course unlike being rich it's generally not considered unethical to remain privileged in that regard.
Perhaps the misconception arises from the authors talking about privilege in specific areas without making it sufficiently clear that they don't mean privilege in absolute terms? Clearly, a gay billionaire is in many ways much less oppressed than a poor straight person. Telling a homeless straight person their oppression is meaningless compared to that of a gay billionaire would be ridiculous. But if we, for example, compare a White registrant with a Black non-registrant, the picture can become much less clear very quickly: A lot of Black children are taught at very young ages to act in specific ways to lower the risk of getting shot by police. Is a constant fear of police brutality since your early childhood a more severe form of oppression than e.g. a 30-year-old who just got registered as a sex offender, but apart from that has had a perfect live so far? A danger of wanting to find an answer of who is "more" discriminated in such muddy comparisons is what some would label oppression olympics, and which usually has a risk of playing into dominant culture's tactic of divide and conquer.
But if one really believes White registrants are much more discriminated than Black non-registrants, and that it's important to convince others of that, then I think showing statistics for each group regarding suicide rates, depression, and getting murdered would be the best way to do so. Otherwise any anecdotal example intented to "prove" that one group is more oppressed is only going to get answered by a counter-example of how the other group being oppressed.
Though the question remains why one would want to show who is "more" oppressed. The post cites Women against Registry as an example of two groups, registrants and women, working together successfully. I might be wrong, but I think instead of spending time on finding out who is "more" oppressed, this alliance instead finds it more beneficial to focus on common goals. I think alliances can work as long as there is a mutual understanding that all groups in a specific alliance experience at least some discrimination, and that it's important to address all of it regardless of its severity.
Though there are some other potential problems with the suggestion of the authors that I see: Do the authors mention if many registrants share their view, or do they just assume they as (presumably) non-registrants would somehow know better than them what is to do about their oppression? And do the authors urge other oppressed groups to become allies of registrants as much as they urge registrants to join other causes?
In any case, if not for strategic alliances, reading about White privilege might help many registrants that so far have been unfamiliar with the topic to gain a better understanding of their own oppression as registrants, why society has developed into hating them so much. Again, one can disagree with some or all of what researchers have written about the topic, but the fact remains that some academics see many connections between how registrants and people of color are oppressed:
"Race, then, may appear to be absent from public and expert discourses about pedophilia, but it is always present, contributing to a much broader ideological project that structures an even wider range of social, economic and political relations. By understanding how normalisation processes draw on racial logics to map out normative and non-normative behaviours, we can see the presence or normalizing discourses as an organizing principle in the very fact of its absence. It is, in short, part of the same tactic of rule that draws on the pedophile and stranger danger to affirm the white heterosexual nuclear family as the normative social unit."
"The whiteness of the 'pedophile' both obscures and serves racism, says Lancaster. 'Shall we say, then, that in a society committed both to a war on crime (with its mass incarceration of black men) and to ridding itself of racism (through formal adherence to a regime of civil rights) the feared figure of the white pedophile is necessary?' he writes. 'Perhaps part of the psychic work he performs is to absolve the guilty conscience of racism at a time when so many other fears are focused on the black gangbanger and brown border menace.' The sex offender, figured white, reassures Americans that they live in a 'post-racial' society."
The War on Sex, page 147