Really nice question, wich is rarely arise.
In fact the answer is I guess :
If you determine the true wave function of your molecule by solving the true hamiltonian of the two well potential + electronic potential, you get the true result which is very close of the results you get using a Lennard-Jones potential.
The Pauli exclusion principle is include in the quantum mechanic !
It is not a fundamental principe of quantum mechanic, it is derived from the commutation of the hamiltonien with the exchange operator $P$ . If the hamiltonian $H$ commute with $P$ then you have two familly of solution : symmetric and anti-symmetric. When solving the hamilotnian for electron you explicitly keep only anti-symetric part. And you can easly demonstrate that anti-symmetric functions follow Pauli exclusion principle !
So it is by explicitly keeping only anti-symmetric solutions that you insure the Pauli Exclusion Principle.
But you can't show ( i'm not sure) that the Lennard-Jones potential describe well this phenomenone. It is totally empiric way but which work