The modern GOP has embraced some very misogynistic politicans and policies. And they pushed Trump into the White House. If you don't want to be labeled a misogynist, don't support a misogynistic party.
It's really that easy. Or have the courage of your convictions and try not to freak out when you live in a place where the majority do not share your views. I say this as a very lefty person in a very red state. I don't hide my views even though I know my neighbors and coworkers may be disapproving.