IMO putting hardwarecloth on the lower part of a chickenwire run is like putting a $80 super-hardened pickproof padlock on a loop of baling twine. Any self-respecting predator will just rip through the chickenwire above your hardwarecloth and away we go.
I would suggest that a better alternative would be something like 2x4 welded wire for the whole run, with something smaller-mesh on the bottom 2-3 (could be hardwarecloth, but could also be 1/2" poultry mesh if you can find it, or even 1" chickenwire with or without the addition of smaller-mesh plastic garden netting depending on how likely you think it is that chickens will fall asleep near the fence or try to stick their heads through). 2x4 mesh is reasonably comparable in price to chcikenwire and will hold up FAR better to assault by a dog. Yes, things can go through or over it, but realistically they can go through or over a chickenwire fence *too* so that is not really a difference from the 2x4 wire.
JMHO, good luck, have fun,
Pat