Ok first thought, which is what solved my problem with my HID projector conversion with the highbeams staying on once activated, is to remove the relay for the fog lights. Don't know why it dawned on me to try pulling the relay when I experienced my problem but I did and voila! it fixed it!
But I would agree that your system does not like not having the resistance of the halogen bulbs. While I am not sure why yours is experiencing this while so many others haven't, my first guess to a fix if the relay trick doesn't work, would be to add a new separate harness like guys do for HIDs or to boost the power to stock halogens or H4 halogen bulb upgrades.