I'd never heard of Laurence Fox before today, so I don't care who he dates. (I have watched his ex-wife, Billie Piper, though. Long-time Doctor Who fan here.) And I'm married, so I'm not looking to date anyone.
But speaking generally about relationships: I do have good friends who differ with me on political and social issues. Mostly, we get along by talking about the interests we have in common, and staying away from the areas of disagreement, and I try not to read their Facebook feeds too often.
For dating, though - no, I don't think I would date an anti-feminist, because gender issues are too integral to dating and marriage. A partner who disagrees with my fiercely feminist views would be a bad match for me; they'd get along much better with a different partner. My spouse and I don't agree on everything in politics -- there's lots of room for negotiation on things like what the minimum wage should be or what are the best gun laws -- but it was important to me that we agreed on gender issues when I decided to marry him.
[ I've changed your "woke women" to "anti-feminists", since you talked about crossing the political aisle. Hope you don't mind.

]