Firstly, the Geonosians were never really Seperatists per see because they were never part of the Republic. They were independent contractors designing and building droid factories for Baktoid who in turn sold said droids to the Trade Federation (who BTW, were also not Separatists, at least on paper.)
It ties into a common misconception that the meeting Kenobi witnessed in AotC was the forming of the Seperatist movement, when in fact it was a business deal between the already established Seperatists (whom Dooku represented) and the various commerce guilds to supply them with arms and equipment and remain officially neutral in the conflict.
Secondly, it's stated in 'Catalyst' that the Geonosians didn't come up with the initial design at all, only refined it.
"The captive Geonosian leader, Poggle the Lesser, maintained that Count Dooku had provided Poggle’s hive with the basic plans, and that the Geonosians had merely refined them."
It's not stated outright but it seems evident that the design ultimately came from Darth Sidious. Where he got it from is still an open question, but given Yoda's line in the Utapau Story Reels about the ancient Sith building superweapons powered by giant kybers one can surmise that it may have been based on an old Sith design. Albeit one lacking the details on how to actually build it, which is why it was handed off to the Geonosian engineers and why it took a genius like Galen to figure out how to make the weapon operational.
So to answer the question: the Seperatists weren't designing the Death Star at all, Darth Tyranus was using his influence as Dooku to have the Geonosians design it for him on behalf of Sidious who, as we all know was playing the long game.