Old garden hose is good for being free, smooth, soft, flexible, easy to cut, long enough pieces, colorful, looks like it belongs in a garden.
Sandwiching the hardware cloth between the bicycle rims and something else would "hide" the edge of the hardware cloth, even if it isn't completely covered, A hoop or band of plastic or metal, which could be pallet banding, strips cut from something plastic like a garbage can (salvage one with a broken bottom?)
Another idea would take the place of the zip ties. Wrap the hardware cloth onto the rim by lacing rope, twine, string trimmer line (the smooth stuff), broken extension cords... through a hole in the mesh, around the rim, through the next hole, and so on until complete. The thicker the stuff, is probably better since the spaces between will be narrower and deeper. This idea for fastening might also be a way of securing the split garden hose.
Some sheet metal thing like a tight U-shape that could slip over the edge of the hardware cloth when flat, and be crimped in place or soldered on, then bent into the cylinder shape together. I'm thinking I've seen something similar done on the edges of lids for fish tanks and terrariums for reptiles. HVAC ductwork is joined using "drive cleats" or "S-cleats" which can be found at the home improvement stores. They are narrow strips that are rolled/folded to have the sharp edges enclosed, and are made to slip the sharp edges of the duct sections into to join them together.