Lighting metal halide lamp
Microchip has a dsPIC app note on electronic ballasts for automotive (not so different from what you have). They use an HV igniter to get the lamp fired. You will also see that AC is used once the lamp is fired. 

A little research showed me that induction lamps don't have electrodes, and need HF applied constantly to excite mercury amalgam into emitting UV, whereas a metal halide lamp relies on an arc to be the UV source, and doesn't appear to have a mercury amalgam pellet at all.

It appears that the induction lamp is specially constructed so that the external electromagnet is close to the amalgam pellet. If there is no pellet in the HID lamp, game over. Even if there were, I don't know if you'd be able to excite it given the dimensions of the glass around it.