#recolinization You need some handwaving. Like, life already existed on the giant’s moons. It has already gone through the steps of developing cells and a DNA code that is optomized for evolution, and formed complex eucaryote cells. Perhaps animals too but they did not survive.
So after the reorganization of the star system, complex cells (eucaryotes) colonized the new Chthonian world. Maybe these were extremophiles living deep in the rocks of the original moon—but that would not evolve animals so rapidly. You need the recolonization to be done with life that already knows to make mu1ticellular organisms. Maybe they were highly evolved biofilms that developed specialized cell types depending on its position in the film, neighbors, and environment. That is, it already knows how to specialize different cell types with the same genitic code.
That’s the general idea. Recolonization took plwce with microorganisms that were able to survive the trip, but were alreqdy “advanced” in the features needed for multicellular plants/animals even if those features had different uses.
#the proposed scenareo
HDE 226868 makes a good point that your proposed planet is contradictory. Let me improve that.
First, it was distant to begin with, so the red giant left the desired result rather than swollowing it. Then, it moved closer to the white dwarf to get warmer.
This kind of relocation involves something like the “grand tack”, as the planet interacted with a new dibris cloud. Look at all the proposed mechanisms for planetary migrations and see what you might get to apply naturally in the aftermath of the red giant phase.
Or, interactions with other massive bodies can rearrange the system.
If you have another star (the companion), why not use it directly? Maybe the gas giant was a planet of the other star, and the two stars are rather close. It is close enough to the red giant to be reduced as indicated. Is it OK for the plot to leave it there?
If not, consider that the planet was originally circumbinary. In the aftermath the orbits of the binary stars were affected (mass loss! Friction from the gas envelope.) and the planet was captured by the new white dwarf in the final stages. Planetary migration mechanisms (involving the companion star too) circularized the orbit.