Okay, war on the cheap: for a war against a small nearby c3 (~6M pop), to conquer, no intention of keeping (no concern for war damage, civilian casualties, etc).
Units used: one Attack & bombing wing loaded with FP, drones, and bombers, and one LRD with 1000 HAV, 1000 jeeps, 120 anti-tank missile bats. Moved in LRD and painted like Mondrian. Took out defense wings with the A&B wing. Took out missile defenses with same--only the minimum needed--I did the capital and 4 of 6 cities. Used ATMB to take out land defenses in those targets. Then HAV and jeeps to take out the targets: undefended pop centers, cap, 3 cities and most of the fourth. Many clicks, well over a thousand.
Cost, as determined by adding up market value of what was lost from the two units, including weapons, ammo, fuel, and military supplies: 568.7B SC$, and 961 personnel. Time: 76 minutes, from the first violation of the c3's borders (by land in this case). Obviously more time in preparation, but that's difficult to compare. Big ticket items: air2air missiles, aircraft fuel, military supplies. I believe it can be done in this manner for slightly less cost, but not much less. Maybe in less than an hour, but not much less. Maybe for much less in an entirely different way.
So for now, I, Jo Jo Hun, the Sixth Musketteer, proud son of the humble soil of Bad Lagerland, claim the modern day record for cheapest c3 capture on Fearless Blue, Walden Two vs The United States of Moressa (now Walden Swamp), on or about Jan. 18, 2580.