While it's nice to protect the fingerboard from wear, it's not possible to insulate it entirely, and it's job is to interface between you and the instrument. Different strings will simply sound different, and even though we will all give you some suggestions, there's nothing like driving there yourself to evaluate. I've seen fretless basses with grooves worn in the fingerboard that still played phenomenally well. It all depends on how you play. Also in the very worst case, fingerboards can be replaced, and it's something we do when needed.
Flatwounds and tapewounds will protect your fingerboard, but they have a softer tone generally. This may be exactly what you're looking for! I think I'm in minority here, as I like the D'Addario Chromes flatwound strings (I use them on my fretted bass lots of the time). They have great flatwound tone, but they are higher tension like a roundwound. If you use a lower tension flawound, like the TI, you will definitely need to adjust the truss rod. If the string doesn't provide enough tension to pull some slight forward bow in the neck, you may need to have a heat bend performed on the bass. If the neck is very thick and rounded, that's more likely as the neck will tend to be more stiff, but again, you can to drive there yourself to find out.
I've also seen people using (gasp!) Rotosounds on fretless basses for an amazing tone. Yes, these will tear up the board over time, but if it's the right sound for you just do what you need to do.
We supply our CX-3 compression wrap strings on all basses, fretted or fretless unless there is a custom request for another string. They are a great general purpose string, and can handle a lot of different techniques.
