A magnet is falling into the coil as shown. The current in the coil creates a magnet as shown ( by the right-hand grip rule ). Now, as the magnet falls towards the coil, the magnetic field strength in the downward direction increases. The induced emf is such that it opposes this change, and so the induced emf decreases the current in the circuit, thus causing the magnetic field strength to decrease to counteract the increase from the magnet. Now, if the power in the circuit decreases, something else must be gaining that energy. But the magnetic attraction between the approaching pole and the pole due to the current has also decreased, so where has this energy gone?
I think your argument is fine up to the sentence:"Now, if the power in the circuit decreases, something else must be gaining that energy." There will be a back emf induced in the circuit by the approaching magnet, and that will indeed reduce the current and the power, that is the rate at which the battery supplies energy. Just because energy isn't being transferred from the battery at as high a rate doesn't mean that energy is going somewhere else!
The falling magnet has a solenoidal electric field (lines of force form loops) which is oriented so that it decreases existing electric current. The induced emf due to that very current decreasing will actually oppose this decrease, but it can't stop it or deny it; the opposing emf can only appear if the current keeps decreasing. This means when the magnet gets close enough, the current actually decreases.
As a result the battery will push less charges and put in less energy per unit time into the circuit (which before got dissipated into heat). The energy that was already in the magnetic field is not lost, it just redistributes in space and part of it may end up as increase in kinetic energy of the magnet.