As demonstrated by the Apollo Command and Service Module's heat shield, it's entirely possible to withstand entry into an Earth-like atmosphere at at least 36,303 feet per second (about 11 km/s or 6.875 miles/second) - i.e. at least from the orbital distance of the Moon. With modern technology, that's probably more, and with the technology your setting apparently has, that's probably a lot more, but "the Moon's orbital distance" is a solid lower bound here.
As such, there's your answer: an altitude of 385,000 kilometers or less and a speed of 11 kilometers per second.
Here's how you do it.
1. Wrap the aircraft in a frame that attaches to multiple points on it so as to distribute structural loads over it evenly. Seal this aircraft/frame assembly in a massive entry capsule, nose-down (i.e. nose facing the heat shield).
2. Once the capsule enters the atmosphere, it deploys an initial set of parachutes and slows down enough that the aircraft inside isn't in danger of getting shredded by aerodynamic forces.
3. The rear of the aeroshell is blown off by explosive bolts.
4. The frame (which is attached to the aircraft) deploys its own parachute - one strong enough to slow the aircraft/frame assembly at the same rate at which the initial parachutes slowed the entire entry capsule - through this gap.
5. The initial set of parachutes, the heat shield, and the rest of the aeroshell fall, are cut, and/or are blown off. This leaves behind the aircraft/frame assembly and its big tail parachute.
6. The frame deploys a smaller parachute towards its nose, dragging the aircraft's nose up. The aircraft is no longer facing directly towards the ground, and is now at an angle the pilot can pull out of more easily.
7. The pilot turns on the engines.
8. The aircraft disconnects from the frame once the engines are fully running.
9. Simultaneously, RATO bottles attached to the aircraft ignite to accelerate it, and the aircraft blasts out of the frame and into level flight.
10. Eventually, the rockets run out of fuel and detach, but, by then, the aircraft's airspeed is high enough to avoid a stall.
11. Eventually, the frame, heat shield, parachutes, aeroshell, rocket bottles and whatnot hit the ground. The aircraft, fortunately, does not.
The angle it launches at is "whatever you want for the sake of the story". Frankly, I have no clue what angle of attack would be the most optimal for these things to rocket off of their airborne frames at.
The entry capsule is going to have to be very big to carry the planes you want to get into the atmosphere, but you said your lander is huge too, so that'll work.
I would have had the aircraft/frame assembly land on the surface before rocket-launching the aircraft, but launching from mid-air gives the pilot more room for failure than launching from the ground. Mid-air, the pilot has:
- more altitude than launching from the ground
- more room to pull out of a dive
- a better launch angle than "straight up"
Also, launching from mid-air lets them deploy over an ocean without needing to attach heavy flotation bags to the frame and risking water damage to aircraft components.
This'd be significantly easier, however, if you were using VTOL aircraft (preferably helicopters) rather than Super Tucanos and F-16s. While they'd still need a heat shield, aeroshell, and whatnot, they could land via parachutes and then take off from the ground, rather than having to "take off" mid-air.
Say...where are these planes going to land once they enter the atmosphere?
