I believe you could build a pretty good 3-axis machine for $3000 (USD, not sure about Looneys), that wouldn't be limited to 2" depth. Adding a 4th axis would probably cost a bit more, but not too much. Going to a 5th axis gets it out of your price range, if only because the CAM software will cost more than your budget. I wouldn't worry about "overkill" - you're probably going to want to cut harder things than foam sooner or later. And the "Joe's" design is somewhat dated at this point; keep reading build logs to find better ones.
If you build your machine from scratch, you can save money, but it requires a fairly well-equipped shop and some skills. Otherwise, think about building from a kit, like one of these: Benchtop CNC Machine Kits | CNCRouterParts or these: CV Supply (205)999-1744