But the whole argument is absurdly bootstrapped.
To assume that women are in a state of false consciousness when they volunteer for sex work, one would also have to assume that all such women have been heavily abused and there is simply no evidence that this might be the case.
Not exactly.
False consciousness is a concept that Rad Fem borrows from Marxism, which is linked to the notion of ideology. According to Marxism and Radical Feminism, our current historical structure of power (defined respectively as capitalism by Marxist and patriarchal by Rad Fem) is made possible by and maintained through both coercion (violence and/or the threat of) and consent.
Consent is produced through the power of ideology, ie, a set of beliefs and ideas which transform a particular historical structure of power (such as unequal class or gender relations) as the necessary order of nature ("this is how it's always been", "there will always be winners and losers", "men are naturally violent and sexual, women are naturally nurturing and emotional", etc.) and the interests of one dominant class (the bourgeoisie for Marxist, men for Rad Fem) into the interests of its subordinate classes (for instance, the idea that workers who took a cut on their paycheck during the last couple years should consider themselve "lucky" because they still have a job, without questioning the fact that being able to provide for one's family should not be considered a privilege, or that those on top of the social pyramid are still making indecent profits that could be otherwise redistributed to pay the workers fairly).
In the case of prostitution, Rad Fem make the circular argument that women cannot freely choose to do sex work, as all sex work is violence and exploitative, and one can't choose to be abused and exploited. Those who claim to be doing sex work by choice are therefore "false conscious", ie, mistaking the interests of the male-class and The Patriarchy (in this case the objectification and commodification of women's bodies and sexuality) for their own.