I was going to comment asking why the ♄-like planet would recede “at a different rate”, but I see question 1 at the end asks us, and you specified science-based. So I'll tell you that no, it's not sensible.
The red giant loses mass because of the high solar wind and even outer layers that can puff off. The planets keep the energy they always had, but now the gravity is less. In a low-eccentricity orbit, the orbit will simply move outward to the new point where the (same) kinetic energy balances the (lower) potential energy by lifting it higher to increase the potential energy again.
All the planets will have a new "groove" for the lower mass star. The change in potential energy with increased distance gets smaller the farther out you go, so the outer planets will move outward more to gain X amount of potential energy.
Ah, here's an idea: the changing situation messes up the mutual stability of the orbits, so the gas giants or anything falling into resonance will throw them around and anything might happen.
The original planet in the habitable zone will be at a larger distance than ours, and the envelope-hydrogen-burning phase will puff out more than ours, too, since you specified a more massive star. You should check the respective sizes to see if the ⊕-like planet will simply be consumed or stay farther in (relative to our case) or what. Likewise, what is the needed distance of the habitable zone during the red giant phase?
The movement of the planets or close approach will not be a huge factor in making the trip feasible. You'll get a launch window repeating just over one year apart.
The deadline might be made by other means: will their planet be engulfed? Maybe the destabilized planet orbits will drive it into a dive or escape. But we're talking millions of years here, so not a rush on the human scale. Perhaps an impending collision will do the trick.
Yea, say they have an asteroid belt too, and not only are the asteroids going to be scattered, but the increasing eccentricity of the planet (due to driving by the giant) will push the perihelion into the asteroid belt. Every year is a gamble against possible major impact events, and the chaos means they can't be accurately forecast. You might need a little hand waving as the reshaping of the orbits are slow processes to get the dire situation set up. Maybe the asteroid belt can be squished to lower its aphelion, too. Hand waving with chaos theory and some buzzwords like strange attractor and phase change in phase space can gloss over that. In any case, invoking a chaotic process makes dramatic outcomes more plausible and any suggested outcome harder to refute.
Point 2: covered above. Not a factor of closeness, but some other deadlines can be brought to bear.
Point 3: more like years, not a month. Transfer orbits have bodies moving the same way as any other orbit: once around takes a year at the inner point, and the other planet's year at the outer point. Roughly figure the time based on the highest point, and you traverse somewhat less than half of it.
For a mass migration, look into a permanent Aldrin Cycler (or several with staggered schedules) transit liner that only needs to be put into its orbit once, and it keeps going around. At each rendezvous you shuttle between it and the planet with light craft.