I'd make sure you're getting a pretty heavy gauge wire if you want to bury it -- even galvanization does not last all that long underground (as soon as the coating is scratched, rust sets in) and you need something thick enough that even when it is largely rusted thru it will still be strong enough to deter a digging dog or whatever.
I would not use anything finer than 2x4 welded wire mesh (like they sell for horse fences) - however, good luck finding it in widths less than 4'. Do I understand that you have already got fencing for the run, and are just looking for something to lie flat on the ground as an apron? If so, in order to get something heavier than hardwarecloth (which I would not trust for long, plus it's expensive) you may have to bite the bullet, buy 4' width, and go along with a heavy wirecutter and snip it in half lengthwise so you have 2' wide strips to use. It would take a While but is certainly possible.
If you
don't already have run fencing up, just get 2x4 welded wire mesh (you might wanna cover the lower part with hardwarecloth or chickenwire) or secondhand chainlink fencing, a couple-few feet taller than you want the run fence to be, and install it with a couple feet of it bent over in a trench to form an integral apron
Maybe someone else can think of sources for 2-3' wide HEAVY gauge fencing, but I can't, sorry.
Another approach is to put down large pavers as an apron, instead. This could be expensive for such a looong fenceline, but if you happen to have a source for secondhand or cheap ones, it's worth considering.
Good luck,
Pat
