These are as much comments as answers, but they do provide some different perspectives.
[This has been edited based on helpful pointers from @uhoh. The earlier comment about the total energy of a test particle being conserved was not correct.]
1. Chladni patterns (sand on a vibrating plate).
In the presence of two large masses in circular orbit, the conserved quantity (akin to energy) is the Jacobi Integral. The Jacobi integral, EJ is the sum of the gravitational potential energy plus the centrifugal potential energy plus the kinetic energy (as seen in the rotating frame). The plots of quasi-potential, EP correspond to the sum of the two potential terms and omit the kinetic energy. Since EJ for a small test particle remains constant (conserved), the kinetic energy (~speed-squared) is the difference between the fixed value of EJ and the quasi-potential surface, EP. So, given a value for EJ for the particle, its speed can be found as sqrt(EJ-EP). Unfortunately this is speed not velocity, so it doesn't help establish the trajectory, but it does tell us something about the probability of finding the particle in a particular place. The probability of finding the particle in an elementary volume of space is inversely proportional to its speed (among other things).
This is the basis through which distinct patterns arise in sand on a vibrating plate, as first detailed by Ernst Chladni. The grains of sand incidentally acquire lateral velocities proportional to the magnitude of the vibrations at a particular point on the plate. These velocities are smallest at nodes in the vibrational modes. The probability of finding grains is greatest where their lateral velocities are lowest - i.e. they spend most of their time at the vibrational nodes (image below).
In general, the motion of our test particle will be chaotic, even with only two large masses present. Particles will have their maximum quasi-potential energy and lowest kinetic energy in the rotating frame when they are close to the L4, L5 points. Lowest kinetic energy means lowest velocity. This means that the particle has highest probability of being found close to L4, L5. This is especially true of particles whose value of EJ is exactly equal to the maximum quasi-energy, EP (which is reaches its maximum at L4, L5). Their speed will slow to zero as they approach the Lagrange points and thus they are almost certain to be found there.
2. Simple Harmonic Motion
For situations where the panetary mass is much smaller than the central mass, the L4, L5 points still exist, but the forces from the planetary mass will be very small. In this case, we can consider the test particle to be in a simple Keplarian orbit around the central mass. Assume the orbit of the planetary mass is circular - as will be the L4,L5 orbits. Assume our test particle is in the same orbit as L4 or L5, but slightly perturbed. We will observe it from a rotating frame attached to L4 or L5.
If the perturbation is made to the plane of the orbit (still circular), then the test particle will appear to oscillate up and down (perpendicular to the plane of the planetary orbit) in simple harmonic motion with the same period as the orbital period.
If the perturbation is made to the eccentricity (still in plane), then the test particle will again appear to move with simple harmonic motion around the Lagrange point. In this case, because of the Coriolis effect, the orbit will be an ellipse with axes in the ratio 2:1 (if the perturbation is larger, the ellipse becomes 'kidney bean shaped').
So, viewed from a rotating frame attached to the Lagrange point, very small perturbations from an L4 or L5 orbit will result in a particle making small elliptical orbits around L4 or L5 with the same period. However, these are not Keplerian orbits. They are orbits with simple harmonic motion such as where the central force is proportional to distance (rather than inverse-square) and the potential is quadratic.
From this perspective, there is no equivalent point mass at the L4, L5 points. Instead, locally, they can be considered to be the center of a sphere of constant frictionless mass density (dark matter!) within which these simple harmonic orbits can persist. The required mass density is the mass of the large central object divided by the volume of a sphere that the planetary orbit would enclose, M/[(4/3)pi.R3]. This ensures that the period of these simple harmonic motions will be the same as the orbital period of the Lagrange points