The spring is likely stainless steel and not particularly magnetic. But it's probably sitting in the trap waiting to be fished out (or cling to your wife's/daughter's hair and cause a snarled mess of hair LOL!)
If you can't get to the back-side of the faucet, you're kind of screwed for either replacing the faucet or retrieving the spring. With our layout, there is an access in the bedroom for the back of the faucet, and the storage compartment underneath that bunk provides access to the trap and drain plumbing.
They definitely don't design these with maintenance and repairs in mind any more!!