Is there a push button on the keyboard, which after switching on you press to initialise the pairing.
On reading your post I tried to pair my own small Bluetooth keyboard, which I occasionally use on my PS3. The iPad 3 recognised it, and it put the message box in the centre of the screen prompting me to input the 4 digit code shown in the box, but by the time I managed to read the code, the box had disappeared and another box popped up with a message saying to make sure the bluetooth device was switched on...hmmm..
I tried it a few times, but no way could I get the code input into the keyboard before the box disappeared with the second error message poping up.
BTW the 4 digit code it asks you to input on the keyboard is different everytime.. so I think there is a compatibilty issue with this particular BT keyboard.
TBH.. I'm not too impressed with Bluetooth devices at the moment, as I've discovered that my Bluetooth stereo headphones won't work with anything being streamed to my iPad 3 as when anything is being streamed it just seems to cripple my wifi, as within a few seconds of watching the streamed TV or whatever, I get a network error message. If I turn off the Headphones it then continues streaming okay. So I'm stuck to using the bluetooth headphones for use just for installed music and films. IOW I'm cannot use them while I'm streaming anything. Apparently it's a commom problem due to both the wireless connection and Bluetooth both using the same 2.4Ghz frequency. I even tried changing different channels and modes on the router, but no solution.
AAMOI, do Bluetooth mice and Keyboards also work on the same 2.4Ghz frequency ? as the wireless uses.
Re. your BT keyboard.. try removing all BT devices again, and persevere, BT devices can sometimes be quarky, ironically this same BT keyboard works okay on my old iPad 1, and it still works fine as I just tried it. I know they have upgraded the Bluetooths hardware on the iPad3, so I'm hoping a few minor iPad3 bugs/ compatibilty issues get fixed in the new iOS update. Apple have already acknowledged there are a few minor glitches with 3g/wifi on the iPad3, so hopefully these will get fixed too.
Sorry I cannot offer a definative solution to your problem, and only mention my experiences above as they are bluetooth related too, I can assure you, your not alone with BT issues...lol