My personal guess is that the GTVA will happily pour money into their military...because that will help the economy via military contracts.
On the other hand, following experiences in the Nebular Campaign and the Battle of Capella, I think the Hecate-class is going to be one of those "This didn't work the way it was supposed to" things. It fends off multiple wings of hostile fightercraft pretty well, but the Hecate lacks the beam firepower to deal with corvette-sized targets (nevermind destroyers) quickly and decisively.
Either they'll turn to a new destroyer design, or they'll upgrade the Orion. I favor the second option at least for the first decade or two. The Orion is already pretty tough; replace the armor with collapsed-core molybendum, tinker with the turret armament, possibly add two or three more small turrets, and you have yourself a pretty good destroyer.
The Hades...the GTVA does have a copy of the technical specifications, minus the sheathe-shielding data, lying around somewhere. (Reference the last Silent Threat CB ani.) They might go for a slightly re-designed version of the Hades that uses mostly or wholly Terran technology. The basic design is certainly sound and can be copied with current GTVA technology.
On the other hand, I think the GTVA will also make an effort to copy the Iceni, which makes a nice substitute for a destroyer if you're not worried about carrying fighters.