We should have been working hard at improving nuclear power, and solving its problems, to the point that this would, by now, be a no-brainer.
The US Navy has been all-in with Nuclear power. R&D has been non-stop. If they haven't "solved its problems", it's unlikely throwing even more money at it, would do so.
the real challenge to commercial nuclear shipping is the operations and maintenance costs; on the nuclear as well as secondary side. The Navy spares no expense in maintaining their fleet, training crews, and keeping large crews to oversee operations; all of which would add tremendously to the cost of a commercial nuke vessel and likely make it noncompetitive with traditional ones. Now, if you could have upscale how power is produced for space probes an run a nuclear-electric ship with a simple nuke side you might have a winner, at least form a technology and cost perspective.