What's the reason for the $5K limit, poor or cheap? Either way, with the European bikes, like the Cagiva, they're cool, but getting parts for one is only going to get more difficult. If you just want a low buy-in, maybe that doesn't matter.
What kind of gravel roads are you thinking about? Smooth and graded? If so, you'd have more options with a full on street bike. Since you mentioned the Ulysses, maybe one of the older tube frame Buells? My VFR is a fuelie, and you can find those around for less than 5. Maybe an FZ6? Older Bandits are wicked cheap as well (don't think those are FI though). Sub-400#s, fuel injected and less than 5 grand isn't possible.
I don't ride, I commute.....