I don't know if what I have to say will help you or if you already know it.
I have the impression that the minimum stable spacing of planetary orbits depends on the masses of the star and the planets and the distances of the orbits..
If the mass of the star is changed, it may change the minimum stable spacing of the planets.
If the planets are less massive, they should be able to have closer stable orbits. For example, tiny asteroids can have orbits with semi major axes which are very close. Hundreds and thousands of asteroids can orbit in a region where only one planet could have a stable orbit.
You also have to consider the width of the circumstellar habitable zones around Alpha Centauri A and Alpha Centauri B.
There have been many attempts in the last 60 years to calculate inner or outer edges, or both, of the Sun's habitable zone. If we know the inner and outer edges, and width, of the Sun's habitable zone, we can multiply or divide it by another star's luminosity relative to the Sun to get the size of that star's habitable zone.
Here is a link to a table with various scientific estimates of the inner or outer edges, or both, of the Sun's habitable zone.
Note how greatly some estimates and calculations vary from others.
The estimate by Dole in 1964 was for planets habitable for humans. I suspect that many other estimates in the list are for planets habitable by carbon based liquid water using life forms, and that some of the inner and outer limits in some estimates may require atmospheres that humans could not breath in to get liquid water temperatures. That is a factor which science fiction writers should research.
So obviously researching the various estimates and calculations and deciding which ones seem most plausible may be a good idea for some writers.
By now astronomers have discovered a number of stars systems with two or more detected planets in stable orbits.
According to one table, the smallest semi-major axis difference between the orbits of consecutive exoplanets is 0.0016 AU or 240,000 kilometers or 149,129 miles, between Kepler-70b and Kepler-70c.
There is so some evidence for a third planet in the Kepler-70 system, orbiting between B and c, which would make the difference between orbits even smaller, if confirmed.
If these planets exist, then the orbits of Kepler-70b and Kepler-70c have 7:10 orbital resonance and have the closest approach between planets of any known planetary system. However, later research3 suggested that what had been detected was not in fact the reflection of light from exoplanets, but star pulsation "visible beyond the cut-off frequency of the star." Further research8 indicated that star pulsation modes were indeed the more likely explanation for the signals found in 2011, and that the two exoplanets probably did not exist.
the smallest semi-major axis difference between the orbits of consecutive exoplanets is about 11 percent between Kepler-36b and Kepler-36c. But the absolute distance between their orbits is larger than in Kepler-70.
Kepler-36b and c have semi-major axes of 0.1153 AU and 0.1283 AU respectively, c is 11% further from star than b.
The difference between the orbits of b and c is 0.013 AU or 1,944,772.3 kilometers, or 1,208,425.5 miles.
among exoplanets known to orbit in the conservative habitable zones of their stars, the smallest differences in orbits are in TRAPPIST-1 d, e, f, & g.
The orbits of the TRAPPIST-1 planetary system are very flat and compact. All seven of TRAPPIST-1's planets orbit much closer than Mercury orbits the Sun. Except for b, they orbit farther than the Galilean satellites do around Jupiter, but closer than most of the other moons of Jupiter. The distance between the orbits of b and c is only 1.6 times the distance between the Earth and the Moon. The planets should appear prominently in each other's skies, in some cases appearing several times larger than the Moon appears from Earth. A year on the closest planet passes in only 1.5 Earth days, while the seventh planet's year passes in only 18.8 days.
TRAPPIST-1d has an orbit with a semi-major axis of 0.02228038 AU, or 3,330,000 kilometers, or 2,069,166 miles.
TRAPPIST-1e has an orbit with a semi-major axis of 0.02928285 AU, or 4.380,000 kilometers, or 2,721,605 miles. Its orbit is 1,050,000 kilometers wider than that of d, or 31.5 percent wider.
TRAPPIST-1f has an orbit with a semi-major axis of 0.03853361 AU, or 5,760,000 kilometers, or 3,769,098 miles. Its orbit is 1,380,000 kilometers wider than that of e, or 31.5 percent wider.
TRAPPIST-1g has an orbit with a semi-major axis of 0.04687692 AU, or 7,010,000 kilometers, or 4,355,812 miles. Its orbit is 1,250,000 kilometers wider than that of f, or 21.7 percent wider.
Of course the star TRAPPIST-1 has a much different mass than that of Alpha Centauri A or Alpha Centauri B, and that may affect how closely it is possible to space planetary orbits.
The PlanetPlanet blog has a section, the Ultimate Solar System, about imaginary solar systems with large numbers of planets in the habitable zones.
In one post it is suggested that 4 planetary orbits would fit into the habitable zone of a star if all planets orbited in the same direction, while 8 planetary orbits would fit in the habitable zone if the planets alternated their orbital directions between prograde and retrograde.
I note that one planet has been confirmed for the Alpha Centauri star system, orbiting Alpha Centauri C or Proximal Centauri. You don't have to worry about such a distant planet messing up the orbits of planets orbiting A and B.
However, it is always possible that planets might be discovered orbiting Alpha Centauri A or B that would mess up the orbits of your planets.
I have heard of computer orbital simulators to calculate the orbits of spacecraft and of astronomical bodies. But I don't know if there are any available to the public that can calculate whether orbits will be stable for billions of years.