Cut 1/2" or 3/8" plywood to exactly fill the cart bottom, place in cart.

You could use more than one piece of plywood, as long as there is no space between them, or between the plywood and the cart sides. 

Set the lift table on the plywood, with equal space on the sides.  Leave 1/4" space between the bolts holding the metal lift table handles in their sockets and the front of the cart.

Screw pieces of 1x2 (or 2/3) to the plywood floor around the lift table wheels to prevent the lift table from moving within the cart.  Use flat head wood screws countersunk in the bottom of the plywood.

Put several layers of duct tape on the front cart edge below the lift table pedal to protect the cart.

The lift table slowly sinks over time.  To prevent the sinking table from damaging the lift table frame (that we'll place on top of the metal top later), cut 4 10" 2x3s, mount them to prop up the lift table.  The props are mounted to the mesh sides with wood screws.  Use pieces of 3/8" plywood between the 2x3s and the mesh and on the outside of the mesh to prevent the gap that would be caused by the frame around the mesh section.

Mount the bumper on the rear end of the cart.