A general problem of DFT, even without the TD part, is that there is no strategy for systematic improvement as you would have it with wavefunction methods where simply throwing another level of unoccupied orbitals into your determinant mix automatically leads to more accuracy at the cost of exploding computation time.

Therefore, everything that's there are benchmark calculations due to which you can say which functional/basis set combination handles what kind of a system well and, expanding from there, you can choose a combination for your desired molecule which handled a sufficiently similar system well enough.

Thank you Dr. Jürgen Weippert

I understand that there is no universal TDDFT strategy that provides accurate results for all molecules. However, I would like to understand the theoretical reasons behind the differences between these methods for this specific molecule. Why does the wB97X-D3 method predict an allowed transition to the second excited state, while the other methods do not? Are there specific factors that enhance the oscillator strength, such as the exact Hartree-Fock exchange ratio?

Allowed or forbidden transition is determined by its dipole moment, high value --> allowed transition, low --> forbidden. The dipole moment in turn is determined by the difference in electron density of the ground and excited states. The value of electron density is sensitive to the functional, but will be even more sensitive to the basis set (the presence of diffusion and polarization components). Check, maybe the matter is in different basis sets, and not functional?