In the ground state of the helium atom, both the electrons have the same quantum numbers $(n,l,m)$ equal to $(1,0,0)$. By Pauli's exclusion principle, the fourth quantum number must be different. What is this fourth quantum number when the electrons are actually in an antisymmetric entangled spin-singlet state?
The spin. Nothing more and nothing less. The fact that the electrons occupy an entangled spin singlet is the natural consequence of the antisymmetrization required by the Pauli exclusion principle when it is applied to two electrons on states that differ only by their magnetic quantum number.