Is the integral action invariant under metric signature convention?

Firstly, note that $S$ does not represent at all the energy of the system. Indeed, $$ [S]=Jcdot s=[ET]. $$ Besides, it contains the Lagrangian but, for a proper definition of the energy, you will need the Hamiltonian and, in general relativity, this is a quite complex and interesting issue.

Then, note that the choice of the signature of the metric is an arbitrary matter and cannot have any impact on the action.

Finally, you can check the action proposed in Weinberg (1972), eq.(12.2.4) and (12.4.2) and Landau and Lifshitz (1951) eq.(93.1) where they postulated identical actions with opposite metrics.