Your problem is in the size of your site. You have huge site and now it’s not enough to have only W3 Total Cache plugin for caching.
During publishing post, WordPress performs a lot of database calls, it updates some information about posts in database. So I can suppose that the weakest spot is in querying to database.
How to solve the issue? It’s very hard to give you one bulletproof solution, which will 100% help you. One suggestion could be in recommendation to try HyperDB as mentioned @brasofilo, but it requires complete understanding in advanced database optimization. Another suggestion could be in hiring a freelancer, who can help you to optimize your site and make it working better.