Calculation of the L4 and L5 Lagrange points don't care about how big the objects are relative to each other. The L4 and L5 points are always at the vertices of equilateral triangles, no matter what the masses of the 2 objects are. The relative masses will have an impact on exactly how stable an object is when placed at the Lagrange point, but not the location of the Lagrange point itself.
True up to a limit. Unfortunately "how stable it is" does indeed depend on the ratio of body sizes, and once that is lower than 24.96:1 it becomes metastable rather than stable.
For comparison Earth:Moon is about 83:1 and Pluto:Charon is 8.2:1 - so Earth-Moon has a stable island at L4/5 and indeed has asteroids sat there; Pluto:Charon does not. A true binary planet we like to get excited about in Canonn has a ratio of 1:1 and definitely does not have a stable L4 point.
The L4/L5 effect arises because the barycentre is a little towards Body 2 from Body 1 - which is not the same thing as being towards the L4/L5 point - so there's an offset in play. Beyond that 24.96 ratio I assume what happens is the barycentre is so far above Body 1 the angle of dangle of that gravity starts pulling in the wrong direction to oppose anything and instead starts reinforcing any drift. I would not like to sit down and work out why it's 24.96, I am scared of arctangents.
The barycentre's position doesn't make a difference to Body 2 (or indeed the L2 point behind it) because in that two-body system there's only one dimension you care about, the barycentre is just moving closer or further away and that just alters what the stable orbital speed happens to be and how far along the same axis L1/L2/L3 will be. But for an observer at L4 that barycentre is indeed moving across the sky as the mass ratios change (
and a little bit closer or farther)
When you perturb Body 3 at the L4/L5 point, there's a bunch of complex orbital mechanics that determine what happens next (we are
in orbit here so it's not as simple as objects going in straight lines in the direction you boot them) but in an L4/L5 system the effective direction of gravity - ie towards that barycentre - opposes the perturbation in useful ways, so when you add up apparent centrifugal, apparent Coriolis, and true gravity from Body 1 plus Body 2 they cancel out.
That extra handy offset vector is not available at L1/L2/L3 because everything's in a straight line, so that's why those points are saddles, not wells.
(Apparent centrifugal and apparent Coriolis are two things that happen because the satellite at L4 IS orbiting something but it is ALSO an object moving in Galilean space and has Galilean inertia, but Newtonian gravity is bending that into a curve; both centrifugal and Coriolis are mathematical conveniences to stop you having to remember mass-inertia is a thing and also that you've been messing around with coordinates a lot; they are both conveniences to allow you to think in points and lines and forget dealing with curves. There's another one called the Euler force which we can fortunately ignore because it arises from angular acceleration, which we don't have here, nobody said nuttin' about spinning these bad boys faster.)
Note that all Lagrange point equations assume that the co-orbits of the two bodies are essentially circular, or near enough to circular to make no practical difference. If the orbits are noticeably elliptical, then that distance between the two objects, at the core of all the calculations, is constantly changing and there can be no fixed stable Lagrange points.
I hate to break this to you but what you get instead is Lagrange orbits.
As Body 1 and Body 2 orbit each other, they still have a stable point at L4, it's just that exact position will move around as the bodies move around. So what you can do is calculate that emergent orbit which is "around" an average L4, and put your spacecraft into an unpowered orbit the same way you would inject it into any other orbit. It's just a bloody weird shape, usually called "kidney bean" and at the centre of it is... nothing.
Either way though the major advantage of using these points is that you can station-keep using
very little propellant and planning for them doesn't tend to rely on any assumption that it would stay there forever if parked. Sussing out the kidney bean orbit is the lowest-energy way to do it.
