Never had a problem like that, myself. Most all HID ballasts generally consume significantly less power than the stock lighting when running, however they do cause a pretty significant spike when first igniting.
The best place to start is probably pulling the cluster out and putting a lead on the power supply wires. See what's going on when you flip on the lights. If there's a big drop below ~10V or so, it's possible that's your issue, as charlie said, a small cap will fix it.
Another thing you can try which is potentially less invasive, is stick a big load on the battery (large cooling fan or some other 12V load of significance) when it's running, and see if that causes your cluster to go crazy as well. If so, you may have battery and/or alternator problems.
I would also check voltage at battery with the engine running, make sure it's over 13V. If not, you might have a wind or two burnt in the alternator, which is not enough to trip the charge idiot light, but will make many things unhappy.