Google Custom Search Plugin conflict with other search plugins

I was tinkering with both Google Custom Search and a third-party plugin (Relevanssi). However, I was unable to get both to work on the same site. The problem seems to be that Google Custom Search hijacks all the search queries – even if they originate from a standard, wordpress search box.

Is there some way around this? I realise that it might be weird to want to do both but Google Custom Search cannot index pages that it’s not allowed access to, whereas I can give admin access to the other search plugins that are installed on my server to index everything.