The distance between stars is such that there is little or nothing along that entire 3 week trip that would need mapping. Space is really, really empty. A more useful conversion on this scale is 1 parsec = 3.26 light years. We're only talking about a 7.5 light year trip here, so the ship can only be expected to encounter one or two systems, assuming a star density roughly equal to that around Earth.
For perspective, even traveling at warp 4, the ship would take about 4 minutes to completely cross a solar system the size of Neptune's orbit. I don't think that it is at all unreasonable to expect that a Galaxy class vessel could map a solar system in 4 minutes. Additionally, there is nothing to suggest that they would be traveling at a constant rate of speed, so if they do want to map a star system more carefully, they could always slow down and spend more time as needed.
From Memory Alpha, Geordi was able to complete a long range sensor scan of a 10 light year radius in a single day. That's vastly more volume than you're talking about.
Your concern about speed is probably too strong. If you're driving down a highway and trying to map out the terrain that is very close to you, it is extremely difficult. It becomes much easier, at any speed, to observe things that are farther away. This is because the more distant objects' angular speed is much lower. The same is true for a jet flying a supersonic speeds. If it is very close to the ground, the trees are moving too fast for a good picture to be taken. If it flies quite high, clear pictures are trivially easy, even at the same speed. For an extreme real life example, mapping satellites travel at several kilometers per second while in orbit but can see whether your kid picked up the toys in your yard or not. For most of its trip, the starship will be quite far from the systems it is mapping, so the ship's speed is more or less irrelevant.