Your #2 option isn't possible, because MusicBee3Settings.ini is saved in your AppData folder which can't be on a separate drive to the program.
(Unless you want to start looking at setting up linked files or other workarounds that aren't worth the hassle)
I would recommend a variation on your #3 option:
* A portable installation of MusicBee on each laptop.
* Your music, library file (i.e. MusicBee database) and playlists on the external drive.
A portable installation makes it easier to keep track of where things are, and the only tradeoff I can think of is that you might not be able to set up file associations for MusicBee in Windows.
It's usually easier to update to new versions too.
You should set each laptop to assign the same drive letter to your external drive each time it is plugged in.
A quick Google search will show instructions on how to do that.
The other reason for moving MusicBee from the external drive is that although you haven't said what type of drive your music is on, if it's over 1TB I'll assume it's mechanical.
The MusicBee interface will be much more responsive if it can read cached album and artist images from a built-in SSD, which your laptops are presumably new enough to have.
It probably won't make a difference where the actual library file is stored, so you could keep that on the same drive as your music for portability and to keep track of play counts etc. across your different laptops.
Though as you've already said, this will require finding some way to replicate other program settings across the different installations if you don't want to update them manually.
But another nice thing about using a portable installation is that you can set one up the way you like it, and then just copy the whole program folder to each of your other laptops.
You can set each copy of MusicBee to use the same playlists folder on your external drive so you don't have to sync them across different installations.
Check Use relative file paths in the Library settings to make them device agnostic.
FWIW, I've been using MusicBee for over a decade, often from a portable mechanical hard drive as in your #1 option, and never had any problems with a corrupted database or settings.
Maybe I've been lucky?
Bee excellent to each other...