Hid Install Issue, No Low Beams

The ballasts have a high voltage generator to strike the arc, so I think it is a good idea to ground the case to prevent interference. The possibility that not grounding them could cause your problem was a long shot in my opinion, but worth a try. The self-tapping screws you described should have been adequate.

If I were a betting man, my money would be on post #8 by "Fiveohwblow". As received in the kit, were the relay module wires (blue, orange, white?) already connected to the headlight connector, or did you have to guess for yourself which one goes where? If the latter is true, then, as Fiveohwblow said, there are 9 possibilities, and the probability of getting it right on the first try without instructions are not much better than 10%.

Incidentally, the factory headlight socket wire colors on my '03 are black for common ground, light green with a black stripe (not yellow) for hi-beams, and white (left side) or blue (right side) with a black stripe for the low beam.

It means the new factory harness connectors are wired correctly.
But the mating HID kit connector could still be wired incorrectly. Did it come pre-wired in the kit?
Do you have a volt-ohmmeter (nothing fancy; one of the $3 Harbor Freight ones will be fine)? You could use the OHMS setting to determine which mating connector wire goes to the 2 HID lamp shutters (which are what determine the high/low beam pattern).

I ran a Google search and discovered several other folks with StangMod HID kit problems similar or identical to yours, including 2 in the 2005-2010 model year forum on this board.
Unfortunately, only one disclosed their solution, which was to rewire the kit connector that mates to the factory headlight connector.

The factory green/black wire probably appears to be at ground because there is a relay coil (fog light cutout) that provides a path to ground. If you use the lowest resistance scale on the meter, you should see a difference between the black, and green/black wire resistance to ground. But the readings on the factory side are not really important.

Remove the cable assembly that came with your kit There is a small 2-pin connector for each lamp assembly (in addition to the 2 connectors that connect to the ballast). This connector is for the shutter. Each pin on that connector should read a low resistance path (less than 1 ohm) to one of the (blue-orange-white) relay module interface wires. The interface wire that does NOT have a low resistance direct path to the 2 pin shutter connector goes to factory low beam. The other two wires should go to ground (black) and hi-beam (green/black) when the kit/factory connectors are mated. You may have to use trial and error to determine which is which, but at least there will only be 2 possibilities. I would test with the battery fuse removed, so the HID bulbs do not turn on. Have an accomplice operate the headlight switches. Both the low-beam and high-beam On positions should cause the relay in the StangMod module to click. Only the high-beam setting should cause the shutter inside the bulbs to move.

These test instructions assume the shutter is designed to operate only for high-beam. You can test this by reconnecting the HIDs but leave the small 2 pin connectors disconnected. The bulbs should still turn on, but the shutters should never operate and both bulbs should remain in low-beam pattern, even when you select high-beam (sort of opposite of the problem you have now).