This happened to me last year in the same area when I went camping with my wife, they don't care if you're doing a bonfire or barbecue, it's just that you can't light fires on the ground/sand.
Hatta, FUJ, RAK, etc. it's usually not a problem (so far). What we've always done is that we wait for the ashes to cool down, pick it up with a shovel, trashbag it and dispose properly along with our other garbage - but I hard time convincing the authorities that's what we do last year because of the other trash-bags (aka humans).
If you go deeper into the desert, I guess you can get away with it, but it's still not advisable due to the environmental impact. Depending on how often you do bonfires and/or camping (we do it quite often in this weather), I suggest you get one of those small portable fire-pits from Ace or one of these.
AED 2,000 is a shitload of money to pay for this type of fine, but now you know! Wouldn't it be nice if the laws were more transparent? Yep. I can also suggest that you visit the Al Madam station and try to get the fine waived (telling them it's your first time, you're new, whatever). Also, the fine isn't linked to your car, as far as I know. I think it's linked to your license, so you can pay for it whenever before you need to renew - don't quote me on that, check up on it.
Good luck!