Will installing a beam under my shed's floor framing eliminate the bounce?
If footings must go below the frost line, then I'd dig them at the edge, one on each side of the building in the middle of the 16 ft long side, and fill with concrete. Then use the truncated pyramid blocks, or other blocks on top to support a beam. Attach the beam loosely on to the joists above, then put jacks under each end and raise the beam (and the shed) until you can slide supporting blocks under the ends of the beam.

The particular footing I use from Lowes or HD has slots at a right angle for 2x lumber and a square for a 4x4 in the center. Here I am using one to support a cedar post (6" x 3.5") so I laid in a piece of PT 2x4 with notches cut out on the corners.