I've got a KLR and two smaller scooters. My opinion is that scooters are more practical as far as storage and they tend to have much better seats that the KLR but the KLR has a better riding position. Riding a scooter with it's more feet forward position and in the case of the Maxi Scooters, more pulled back handle bars may bother your back. If possible take a Burgman for a long ride and see how your back feels.
My recommendation, keep the KLR AND
get a scooter. One bike is just not enough