Stone, have you tried your smother without the carpet? Nothing I've tried to smother under cardboard (corrugated box kind, not cereal boxes) topped with leaves/mulch has failed to die.
It has to overlap enough to prevent exploitation of the seams, have enough material on top of it that is thick & heavy enough to cause the cardboard to conform to the soil contours, block the light, and be left undisturbed until death is complete. Severing around the edges, down to about 5", especially if dealing with creeping grass, seems to be very important to successfully creating a definite edge where the grass stops. Punctures from rough material will thwart the efforts, allowing growth to exploit the holes, so it's also important to make sure there are no sticks, pine cones, stumps of mowed entities with woody stems to cause punctures under the cardboard. Does not need to be removed to use the bed (once weed mortality is confirmed,) and helps improve fertility, tilth, drainage of soil. Can be augmented/renewed without disturbing existing material.
In the particular instance asked about, I would do this on the inside of the fence/cage, and lay bricks or timbers along the outside edge, something you can use weed trimmer against, so nothing crawls over the top. That would be easier against the flat edge vs. the mesh of the fence/cage. If the thick cover over the inside smother (leaves, mulch, weed material pulled before making seeds, any available OM you want to use) is maintained, nothing is going to grow from under. Sprouts are easy to spot, easy to pull.
