Another crackpot idea:
Instead of a catapult or a flying battery pack, go for a tethered launch. The plane taxis onto a platform that is actually a powerful "cable laying UAV" with powerful ducted fans for propulsion and some lift and two big powered spools of copper as heavy as it can carry. The platform arrests the plane's main undercarriage, connects power lines to the plane and raises itself on its own set of wheels. They accelerate together, leave the ground together, all while the powered spools are unrolling the wires to the ground station exactly as fast as needed to avoid mechanical drag (the wires are effectively in free-fall, perhaps issues with the cable landing could be evaded by sending up a chain of "cable carrying (T)UAVs" that take the role of virtual poles). In time before the spools run out, the platform disconnects from the plane slows down the spools so that the inertia and main motors of the platform start pulling on the wires, then reels itself back in while it maneuvers to a point above the base station, finishing the circle with a tail-landing followed by dropping into "platform" position controlled by its secondary fans. Meanwhile, the plane flies of into the sunset on a fresh set of batteries.
But the battery tender UAV actually sounds far more practical, I guess I just like ideas involving tethered UAV.
(and somehow I feel almost entitled do spew out the most improbable ideas, since seeing footage of the successful dual suicide burns of the Falcon Heavy boosters - if that's possible, why isn't everything else as well?)