The Specter terminal clearly states that the reaper war and crucible project would bankrupt the galaxy within a year, there is no way an ark project would be able to done along side the crucible.
The Spectre terminal message never mentions the Crucible. It only mentions military spending, without going into specifics on expenditures:
This financial report is for top-level embassy access only. According to financial estimates from First Irune Investment Trusts, a division of Elkoss Combine, the Citadel races cannot sustain the cost of the war effort for more than one year. Although military spending has stimulated certain market sectors, shortages and rationing in other areas has depressed the financial system. Civilian discretionary spending has fallen except in entertainment areas like escapist vids or drinking. First Irune is borrowing heavily to provide funds for key functions across the Citadel, but if the war does not end within the next year, galactic financial collapse is inevitable.
An ark project is going to have thousands and thousands of people involved it'd be easily infiltrated by Cerberus, STG and the shadow brokers agents. Since Liara is the shadow broker by ME3 she would've learned of the projects existence.
Not necessarily.
The Manhattan Project would eventually involve around 130,000 people. It remained a secret throughout much of the war including to high level personnel within the U.S. government and military.
Actually the fact that Cerberus and the Shadow Broker seemed to be omniscient was itself a bit of a stretch. Even the most efficient intelligence agencies in the real world, never know entirely what rival nations are up to. It would be somewhat realistic if the Council managed to keep something secret.