Another major issue with NSWRs is propellant storage. This isn't stuff you want puddling up under a leaking tank. Especially when you consider that you're a walking neutron reflector.
A while back I came up with a potential way to use fusion in a NSWR...the Antimatter-Catalyzed Fusion-Boosted Nuclear Saltwater Rocket, or ACFBNSWR. Not only would the fusion contribute directly, with fusion as a source of neutrons your fissile saltwater doesn't need to be as highly enriched, and in fact needn't be capable of sustaining a reaction on its own...better safety through antimatter!
As for fission fragment engines, they would only spend a small fraction of their operational lifetimes in Earth's vicinity even if not initially boosted away from Earth on chemical rockets to avoid a slow outward spiral. Away from Earth, apart from being diluted to the point where you'd be lucky to ever encounter a single fragment, I don't see why they would stick around when the charged particles of the solar wind don't. In a perfect vacuum they might be trapped by the sun's magnetic field (the gyroradius is around 47000 km in the area of Earth's solar orbit, if I crunched the numbers right), but I suspect they'd instead flow outward with the solar wind, possibly with their trajectories curling around approximately that radius until they're headed outward with the wind. I'm no plasma physicist though.