I realize this question is old but perhaps my answer can help someone else who comes across this post while searching for a solution as I did.
What I ended up doing was the following:
Get some ‘1 3/8” panel clamps’. I bought mine from Home Depot. These are also called “kennel clamps”. They will hold two 1 3/8” pieces of tubing parallel to each other. You will need one clamp for the top and one for the bottom.
Get some ‘1 3/8” post caps’. I got mine off Amazon, since Home Depot didn’t have any. You will need one cap each for the top and bottom of the tubing you will use to fill the gap. This will keep water and nesting wasps, etc., out of the tubing.
Get some 1 3/8” chain link fencing “top rail”. In my case I had some left over from a prior project.
Measure the length of the gap beneath the gate latch to the bottom of the gate. Be sure to leave a reasonable amount of space beneath the latch so there is room for the post caps and things don’t end up looking like they are crowded together.
Cut a length of the top rail according to your measurement. I used an abrasive cut-off saw because I have one, but a hacksaw or even a Dremel tool with cutting wheel and patience will do. Put one of the post caps on either end of the tubing.
Fasten the length of tubing to the side of the gate using the panel clamps, suspended between the gate and the terminal post beneath the latch.
Once assembled, the new filler post will move with the gate. It even looks professional.
In my case, my fencing is black powder-coated. None of the materials I was able to purchase quickly were black, so I spray painted everything with two heavy coats of black gloss paint before assembly. It looks good.
This was done in my case to prevent my wife’s rather devious Scotty dog puppy from getting out through the gap.