May the Government validly invoke the doctrine of State Immunity from suit if its invocation will serve as an instrument for perpetrating an injustice on a citizen?

No. The doctrine of governmental immunity from suit cannot serve as an instrument for perpetrating an injustice on a citizen. Ministerio vs. CFI of Cebu, G.R. No. L-31635, August 31, 1971, 40 SCRA 464 

In EPG Construction vs. Vigilar, G.R. No. 131544, March 16, 20013, 54 SCRA 566, the SC held that as the staunch guardian of the citizen's rights and and welfare, it cannot sanction an injustice so patent on its face, and allow itself to be an instrument in the perpetration thereof. Justice and equity sternly demand that the States cloak of invincibility against suit be shred in this particular instance, and that petitioners contractors be duly compensated on the basis of quantum meruit for construction done on the public works housing project.