I agree that raising the drop ceiling in the area outside the duct is the right move here. However, if you insist on hanging drywall, you have a few options.
Suspend furring strips. You wouldn't use a full-height strip attached directly to the joist. You'd use slats mounted to the joist sides to attach to the furring strip sides. Something like this:
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The horizontal members must be up to the task of carrying drywall for the span between supports. Be sure to use an approved attachment method to the joists. Keep all fasteners in shear (horizontal), and use enough of adequate size for the load.
Or look for a commercial solution. There are systems similar to suspended ceiling track designed to carry drywall. You install them and screw the drywall to the channel rails.