I can't provide the exact cost of movement replacement at Cartier but I estimate it would be several hundred dollars.
I don't know if the Cartier 81 movement is interchangeable with an ETA calibre. If it is, then replacement would be much cheaper, but then swapping the movement is not officially sanctioned and Cartier will no longer service the watch afterwards as it would be a "franken" watch. Naturally the Cartier boutique would not advise on an ETA replacement.
But personally speaking since the watch is inexpensive in the first place, if you can get your watchmaker to swap the movement you'll have a working watch at low cost. I assume your daughter doesn't really care if the movement is original or not, as long as she has a working watch.
- SJX