I just wonder how much of that was Sidious' influence, given that he and Dooku took over the commission of the GAR.
Well, yes, naturally -- the whole reason Palpatine/Sidious orchestrated the Clone Wars was to give himself an excuse to seize ever greater power, and that included creating a justification for the manufacture of an enormous warfleet. Although, of course, he drew on resources from both sides in the wars -- things like Star Destroyers and TIEs and walkers were the descendants of Republic hardware, and Stormtroopers were an outgrowth of Clone Troopers, but Palpatine commissioned the Geonosians (on the Separatist side) to design the Death Star.
But Dooku wasn't behind the Republic army, he led the Separatist forces. Palpatine set Dooku up in the role of the enemy that he put on a public pretense of fighting against as Chancellor. (I'm not sure if Dooku even knew that "Lord Sidious" was really Palpatine, given that Sidious always appeared to him hooded.)