Edelweis works more like a charter airline and hence has their own logic - you need to book directly with them:
You can book seats here:
B2S (flyedelweiss.com)
You do NOT need a booking code. You need your exact flight dates and times, and after having booked your seat you need to put in your names that are IDENTICAL to the one in your booking with Swiss.
(Don't worry if the flight numbers are not identical If you have LX8XXX this will the same as WKXXX
You should be able to book business seats. Just go ahead and book your seats. Business will be for free, Eco will come at a charge.
(I have done so many times and works like charm....)
More info here
Seat reservation (flyedelweiss.com)
And yes I am always amazed that for seat reservations and also for checkin with Edelweiss you only need your name and nothing else....
(Altough if you had booked a Swiss Flight operated by Edelweiss you sometimes need to check in with Swiss and not Edelweiss directly, and then you need your Swiss Reservation number, this depends on the airport)
Happy travel on on Edelweiss