How do I set an Interactive Designer template to have a color picker available as a background color option?

Edit your template and go to Template Info > Background Page Options. Make sure Allow solid color as background is checked and check Enable RGB Color Picker and/or Enable CMYK Color Picker
Then under Per-Page Options, select Default for all pages, or select the page the background color picker should be available on, and hit Set Page Options. A confirmation popup will be shown after hitting Set Page Options, click OK. Then click Save Background Files Options
When you test the template, there should be a Background Options control showing on the left, using the Choose button on this control will show the color picker.