Using wire mesh on yard gate for exclusion
Instead of hardware cloth that has to buckle and flex to cover the gap, consider a thin stiff material that could be fixed to narrow the gap. I don't know what size of rabbit skulls you're up against but I have to imagine that if you can narrow the gap to under an inch from the 1-3/8 it is now (ie, block off 1/4 inch from each side) then the heads, and the rest of the bodies, won't be able to pass through.
A bit of material installed to the stile and the jamb (can I use those words in the context of a gate?), in line with the hinges, will narrow the gap while the gate is closed but won't bind or prevent the gate opening either direction. Here are some materials you might find in a home center:

1/4 inch square stock (steel or aluminum)
L profile &quot;angle iron&quot; with legs around 1/2 inch
sheet metal folded to an L shape (cut down pieces of rectangle HVAC duct)
flashing/trim/etc from the roofing department

Fasten with screws, magnets, adhesive, foam mounting tape, or whatever suits you and the chosen materials.