Moving to the center on social issues won't help the GOP. It'll just send their white working class voters into the hands of the Democrats, who actually would do a better job representing them anyway, at least on pocketbook issues.
How many white working-class Republicans think like that though? "Oh I'd vote for the Democrats, except I'm a social conservative." In my experience, people who are right-wing on social issues are also usually pretty damn right-wing on economics, regardless of income (this is especially true in places like the South or the West). And as opebo often alludes to, a lot of the economic AND social conservatism of working-class and working-poor whites is due to racism (and also paired with sexism, particularly among white men, and xenophobia/nativism regarding immigrants).