You need some handwaving, such as life already existing on the giant’s moons. It has already gone through the steps of developing cells and a DNA code that is optimized 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 multicellular 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 genetic code.
That’s the general idea. Recolonization took place with microorganisms that were able to survive the trip, but were already “advanced” in the features needed for multicellular plants/animals, even if those features had different uses.
The Proposed Scenario
HDE 226868 makes a good point that your proposed planet is contradictory. Let me try to improve on that:
First, it was distant to begin with, so the red giant left the desired result rather than swallowing 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 debris 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. Another possibility is that 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 and 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.