Custom colour swatches in Customize panel

As we have a brand colour palette at work that we have to conform to, I’m wondering if and how I can modify the Customizer to replace the colour picker in the colours section with custom swatches.

I’ve been following the tutorial at http://www.peachpit.com/articles/article.aspx?p=1918339 and have it working with the built in colour picker, but can’t work out how to change it to use swatches instead. We don’t really want to offer the colour picker as people invariably pick non-brand colours.

My code so far is virtually identical to the tutorial: http://pastebin.com/Y74KVAs6

Can anyone here explain (in simple terms – my understanding of php is basic) how I can do this please?