Glad you mentioned intersectionality, and particularly Crenshaw-Williams' concept of it. It recognises that there isn't a simple "scientific" formula of privilege and oppression (i.e. you don't get a plus point for being white and a minus point for being female, or any variation on that theme,) but that there can be a unique experience for individuals whose identity IS at an intersection between 2 or more identities.
In my experience, feminism, probably more so than the political ideologies of those who advocate for the rights of other oppressed groups, tend to be more inclusive in their definition and less likely to buy into a hierarchy of oppressions (i.e. it's worse being female than being black, it's worse being Lesbian than being gay, etc.)
So, you can be an advocate for women's rights and for workers rights, you can be an activist for women's rights and an anti-racist campaigner, etc. etc. without conflict.
Having said that, some feminists can struggle with "wearing" the privileges that some women enjoy while others don't. I think that can be the case for anyone who fights for the rights of any disadvantaged group where inevitably some within that group will still have more privileges than others, due to things like ethnicity, language, gender, sexual orientation, etc.
So yes, working class women can experience the impact of patriarchal institutions, traditions, practices, etc. in a different way than middle and upper class women do. For example, working class women may be unable to take an extended period of maternity leave from work if they need to get back to their full earning potential as soon as possible. Middle class women who had the benefits of a good education may have more earning potential if their relationship breaks down than a working class woman who's only done manual labour.
However, regardless of class, education or other privileges, women can still be raped by men, can still be abused by partners, can still be denied equality of opportunity in employment, are still judged on the basis of their appearance rather than their achievements in society, etc.