First of all, I would recommend calming down. The money you could have saved is not worth a nervous breakdown.
The good news is that you got $4500 off a vehicle that performance wise is as good as the 48i. The bad news is that you could have probably negotiated down another $3k down. When you look at invoice, you have to factor in all the features that comes with the vehicle so don't compare it with $47k.
Your overall savings off the MSRP was $5k which in itself is not bad. I purchased a 2010 X5 3.0i with $2.5k holiday credit and an additional $3.7k off the sticker. It didn't come easy though, I had shop by dealer and get the best deal before ever entering their doors.
If you already signed the paperwork, I would say take a look at the fine print and see if you have a window to return the car if you are not satisfied. You can then use that to reopen negotiations (don't threaten though, be cool and firm about it). If there is nothing in the fine print that allows you to return the vehicle, you can call the dealership posing as a new customer (or have your wife do it) and ask the question if whether after purchasing the car you have a window of time to return it if you are not satisfied with it. They will likely give you their most flexible answer because they see the potential of a new customer. You can then use what you are quoted to go back in to return or renegotiate.
Worst case, don't fret it, you didn't save as much as you could have but it was not a bad deal.