You can take a sterile needle and empty syringe, insert it into the center of the swollen area of one cheek, and draw the plunger back. If there is fluid - blood, pus, anything - it will collect in the syringe, and you won't have an open cut like with lancing. If there's no fluid you can still do a "needle biopsy" by drawing a small sample of flesh. So if you collect anything maybe you could have it tested at a vet's lab? Even if they don't want to see the duck they may be able and willing to do some lab work for you, and then you'd know exactly what antibiotics would kick that thing's butt.
GOOD LORD...