In a multi-website setup, we are currently storing a value in an anonymous user session if the user has entered something on a form. This is just used to redirect them on a few of the websites to a specific page. What we've found is that if full-page caching is enabled, then if any anonymous user has entered that form value, then all users aren't redirected, they're instead shown the normal content. I think this makes sense based on how full page caching works.
My question is, is it possible to check an anonymous users session for that value / redirect them with full page caching enabled? Right now the performance hit without it enabled is fairly large (but things work correctly once it's disabled). If that's not possible, is it possible to turn off full page caching on specific websites, but not all of them?