WordPress is very cookie heavy. I don't think Varnish OTB would work in front of it. Sure you might be able to configure it to work, but that's probably more effort than just tweaking a few lines of Apache.

I don't think WordPress really uses cookies unless you're logging in or posting comments. As long as most of your visitors are read-only (which is usually the case, especially under heavy traffic), they won't get cookied.