Note: Answering from a comment posted on Space Exploration
The classical stability analysis of these libration points assumes that we are examining the motion of a particle whose dynamics are perturbed by the gravitational impacts of a primary and secondary mass, so as a bottom-line-up-front type of answer, the mass of T is negligible - so any large increases in mass will negate these assumptions. Further, the stability analysis is linear stability analysis, implying that the stability is only valid within a neighborhood of the equilibrium point, and very little information can be said about the non-linear behavior (However, an unstable equilibrium point will be unstable in the non-linear dynamics).
With that said, the critical mass value in the circular restricted three body problem (CR3BP) can be found from the following development, summarized from most major astrodynamics texts to include Vallado (1), Roy (2), Schaub (3), or the essential 1967 CR3BP text by Szebehely (4). The linear variational equations of motion for small in-plane perturbations about the triangular libration points can be found as
$\ddot{\xi}=2\dot{\eta}+U_{xx}^*\xi + U_{xy}^{*}\eta\\
\ddot{\eta}=-2\dot{\xi}+U_{yx}^{*}\xi+U^{*}_{yy}\eta$
where $\xi,\eta$ are the perturbations in the $x$ and $y$ directions in the CR3BP synodic frame, and $U^{*}_{..}$ is are partials of an artificial pseudo-potential function. Essentially, the characteristic equation for this linear system is found as $\Lambda^2 + \Lambda + \frac{27}{4}\mu(1-\mu)=0$, where $\Lambda = \lambda^2$, $\lambda$ being an eigenvalue of the actual characteristic equation.
If we let $g = 1-27\mu(1-\mu)$, the four roots of the system can be expressed as slightly complicated functions of $g$, but the eigenvalue behavior can be classified according to the value of $g$ as below:
- $0 < g \leq 1$: Pure imaginary eigenvalues, marginal stability
- $g = 0$: Repeated eigenvalues; secular terms present; unstable
- $g > 0$: Eigenvalues with positive reals; unstable
The critical $\mu$ value ($\mu_c$) comes from setting $g=0$. Solving this, we find that $\mu_c=\frac{1}{2}\left(1\pm\frac{\sqrt{69}}{9}\right)\approx0.0385$. Again, a key assumption in this development is that the mass of the third body is negligible. A lot of systems of interest are below this critical mass value to include the Earth-Moon, Sun-Earth, Sun-Jupiter, etc.; however, some systems are definitely above this value -- consider the Pluto-Charon system with a $\mu$ value of approximately 0.1101.
1: Vallado, D.A. Fundamentals of Astrodynamics and Applications. 30 Jun, 2001. Springer Science & Business Media.
2: Roy, A.E. Orbital Motion, 4th Ed. 31 Dec 2004. CRC Press.
3: Schaub, H.P. Analytical Mechanics of Space Systems. 2003. AIAA.
4: Szebehely, V.G. Theory of Orbits in the Restricted Problem of Three Bodies. June 1967. Academic Pr.
