Perhaps you could find a bed frame without a headboard or footboard, and then get a long bed skirt and comforter, so it LOOKS like she has no frame? Maybe one with drawers underneath, even?
Ack!!
I have a Malm that a friend gave to me because he got tired of the bottom suspension falling through and dropping the mattress onto the floor. I brought it home, re-engineered the mount for the center bracket, reinforced all of the connections and haven’t had a problem with it since.
I vote for only a sleeping bag on the floor if she balks at the frame.
Another vote for letting her have the box spring on the floor.
That said, if you want there to be consequences, you could insist that she buys the rug/whatever floor protector to go underneath it to stop the floor getting scratched. That way she gets her choice but not without a consequence for breaking the frame.