We have carpet on our private course and it works great. You do need to grade out the ground below it very level; imperfections that you wouldn't mind on an earth tee seem magnified under carpet, perhaps because you feel but can't see them.
Staking down the back corners is important, to prevent slides. Even a half-inch slide will mess up your drive.
Be careful with low areas or places where rainwater might flow. One drawback to carpets is that, when they get soaking wet, they take a long time to dry (worse with indoor carpet). And if mud can wash onto them, it gets in them and they're slick forever.
Finally, we tried indoor carpet, upside down. It provided terrific grip. It also deteriorated quite quickly.