My dexter had to be on liquid doxicyclin last year for Lyme's disease. Although they make it beef & cheese flavor (ewww!) he hated it. The vet said to get the dropper as far back into his mouth as I could, past his tongue and he'd have no choice but to swallow because it would be behind the part of his tongue that could get it back up.
Prayers to you and Gizmo!