I am stuck with figuring out what to do with the existential quantifier in the third premise.

Let us have a quick look. You are aiming to derive Gse from the premises, so first use universal elimination on the third, and then...

 | Pd ⟷ (Hj & Mj)
 | Gsd
 | ∀x∀y∃z(((Gxy & (Py ➝ Pz)) & Rxyz) ➝ Gxz)
 | Pe ⟷ ∀x(Hx ➝ Mx)
 | Rsde 
 |- 
 | ∀y∃z(((Gsy & (Py ➝ Pz)) & Rsyz) ➝ Gsz)
 | ∃z(((Gsd & (Pd ➝ Pz)) & Rsdz) ➝ Gsz)

… Nope, you cannot do anything else. You cannot establish that term e is a witness for that existential, and further you cannot derive that Pd ➝ Pe . You have no route to deriving Gse ; it simply is not entailed by these premises.