Automated, parallelized, cross-browser testing is a seductive idea, but in my experience, the cost may not justify the benefit. I do not know whether your customer requests are based on actual experience or the advice of a trusted professional or the unfounded claims of someone in an elevator.
You may spend a lot of time maintaining your cross-browser tests. Even if you do not change your product, today's tests may work with today's browsers and break with next week's browsers. If you decide to leverage Selenium Grid, expect to spend a significant amount of time upgrading to new versions of Selenium and diagnosing whether the latest bug is in your test, in Selenium, or in your product. (Of course the bug may also be in the browser, but that is outside of your control.)
As @Siva mentioned, even if your functional tests run perfectly, you still have not verified browser-compatibility. No matter what, you still need to test manually.
You have a limited testing budget. The time you spend maintaining your cross-browser tests is time you cannot spend testing in other ways.
Still, your experience may be different from mine. I recommend approaching automated cross-browser tests as a limited-scope pilot project. Automate just part of the app and observe how much effort you spend keeping it running as the product changes and new browser versions are released. If you are satisfied with the expense, you can leverage technology more aggressively; or if you decide the benefit just not justify the cost, you can throw it away and think about something else.