The potential cost saving isn't really the issue here. And for one off keys the saving is not that big from a mobile (or 3rd party) locksmith.
A mobile locksmith would charge typically between £100-£150.
I can do them for about £20-£25 ea for my own cars but I don't offer it as a service since the non-BMW way involves removing the EWS, taking the circuit board out and attaching a CPU clamp to the board to hack it. If I bust my EWS doing it hard luck. If I bust someone else's I don't want the responsibility. I don't have the time. And for the amount of agro I would want £60+ anyway.
Now if you had the key you would at least have the option of going to a mobile (or other independent) locksmith to have the key cut from tracing and the transponder programmed to the car. That programming is what the EWS (the immobiliser) checks to let starting go ahead. Synching the remote is easy and has nothing to do with the security of the car. People get very confused about this!
A dumb key will open a lock or turn the ignition barrel. Only a fully programmed key will start it. Synching will not and don't let anyone tell you this! In fact the key doesn't even need to be synched to start the car.
Here's the ultimate problem. And it's not that you don't physically have a key any more. It's that you don't know the code of the key. Instead of tracing the key, a computerised cutter can be fed the key code.
Every key has a code associated with it. There are commonly 2 ways of discovering the code.
A skilled locksmith can 'read' the code purely from a picture of the key. And that's how I get my keys physically cut. I've got at least 2 keys for every car in case I lose one. And I have records of all the key codes. And backups of the EWS from each of my cars.
Or I can only assume that a BMW dealer has a note of your key code somewhere in their system. But don't 100% rely on that!
Since you don't presumably have the key code (or a picture of the key) then the only alternative to the dealer is to see if a locksmith can dismantle the door lock and read the code from the individual tumblers. Or quite possibly use a special tool which doesn't involve dismantling - I know how it works but I'm not going to go into here for security reasons. I'd lay odds they could do either but it's whether they would want to and just as importantly if it's cost effective.
In short I would go and see your BMW dealer. He will undoubtedly want ID so V5, passport, utility bill etc. And quite possibly your inside leg measurement and what colour pants you are wearing

And hope they have the key code.
Did you tell the BMW dealer you didn't have a key? Or did you just ask for the price of 'a key'.