Really boils down to where you need to be, how much you can afford, and how much time you want to spend in traffic.
Each of the neighborhoods in Seattle have a very distinctive feel, and it might be hard to see that in such a short period of time for your tour visit. Also note: public transportation in Seattle straight up blows compared to what you're used to. I'll comment based on what you wrote:
Ballard: I lived here for a few years and really enjoyed it- this was about 15 years ago when it was filled with "the newly wed and the nearly dead". It's only become more hip and popular since. Pretty white-washed without a lot of diversity, unless you consider the rivalry between the Sons of Norway and the Swedish Council. Getting in and out of Ballard just plain sucks since there is two neighborhoods between it and the main artery of I5. Getting downtown from Ballard is one decent thing though. You can bomb down 15th right into the waterfront of downtown in a few minutes.
Fremont: Slightly closer to downtown than Ballard. Fremont has prided itself on its weirdness for years, and it continues to be a pretty neat area. Used to be kinda rough industrial and fishing areas, now pretty similar to Ballard with a lot of younger families. Similar access to downtown, which is to say not great, but better than a lot of the outlying areas.
SLU: Do some reading on SLU and the general rage of longtime residents against all things Amazon, and you'll have a good idea about SLU. This was nothing more than an industrial park up until about a decade ago, now it is one of the fastest growing areas. Not a great deal of continuity or feel of consistency in community here, but that's a new neighborhood for you. You're close to everything, but in the heart of the worst surface street traffic in the whole city.
Finally, Capitol Hill. Cap Hill is proud to be the home of all the freaks in Seattle, which should translate to mean that it is a pretty neat spot. Huge LBGTQ community, and generally lots to see and do. The theme on Broadway is generally pretty garish, but it grows on you fast. One of the easiest spots to get in and out of, but higher prices are the result.
If you need to get to Pioneer Square, then you might also consider West Seattle. Great neighborhood, and successfully feels like it isn't part of a big city. Just out of the way enough to feel like its own town. I really liked WS. Good opportunities for lower rent/home prices here too.