For short travel times like this you could use very refined solar sails (theoretically an advanced sail could go from the Sun to Pluto in as little as 3 years) or massively powerful fusion drives. Either way you are now looking at extinction level events moving through the Solar System, since the kinetic energy of ships moving at such speeds makes nuclear weapons somewhat trivial.
This would make people on the terraformed worlds very nervous, since accidents that cripple the drive or navigation system mean there is a possibility of something arriving on a collision course with gigatonnes of energy.
Based on your conditions, I could assume that initial settlement was by more conventional ships taking years to arrive (still moving very rapidly in terms of kinetic energy, but "only" in the kilotonne to megatonne range), so settlers have been aware of the potential danger since "day one" 200 years ago. As technology is refined and higher velocities become possible, I suspect that every inhabited body in the Solar System will take effective steps to prevent or minimize the possibility of high energy impacts. Essentially, there would be a solar system wide "ATC" to control the arrival and departues of spacecraft (no "tramp" ships anywhere, ever), and a powerful "Space Guard" or even Space Navy capable of vaporizing any potential threats to inhabited colonies.
Colonies themselves would be heavily built (initially to be sealed off from hostile environments and provide radiation protection), and I could see a very paranoid civilizations arising, with colonies being built from multiple small shelters which can independently survive should a nearby strike or other event cripple parts of the colony.
So we have billions of very paranoid and cautious people inhabiting the Solar System, looking grimly skyward as ships approach on high energy orbits, keeping their fingers hovering over the triggers of a planetary defense system, just in case....not exactly the environment for large scale peace and collaboration. The Solar System could accidentally become a very restrictive and tightly controlled place.