I have a funny feeling somebody used the wrong switch. I believe what you do manually, is what the switch does.

So, replace the switch with one designed to turn on a florescent light.

The desk lamp I have with that starting system is a single tube. Eons ago, the bas got painted and a new switch installed.

See http://www.edisontechcenter.org/Fluorescent.html for operation.

Here's another place to look. http://www.egaynor.com/edwin_gaynor_lampholders/666.htm These use two ballasts for a two bulb system.