What would be the fate of a second satellite at the mid-point between the Earth and the Moon, with half the mass of the Moon?

If this hypothetical moon were to come to exist between the Earth and Moon, how long would it survive? And could it transition to the moon's orbit, as a subsatellite?

To clairify, this scenario is intended to occur after a second giant-impact event, resulting in a second, smaller moon capable of creating a bi-lunar(?) eclipse, albeit with the new moon presumably colliding with Earth or the Moon after an extended period of time.

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What would be the fate of a second satellite at the mid-point between the Earth and the Moon, with half the mass of the Moon?

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