Jules wrote:

I'm thinking 4x4" posts and pressure-treated 2x4s, with three rails -
but putting the bottom-most rail as close to the ground as possible
(that way I can fasten the mesh to it without needing to dig it into
the ground, but I can always bury cheap chicken wire right at the
bottom later if the dogs work out how to dig underneath).

Put a post a foot or so from the house, and instead of having the rail
terminate at the post (like the other rails), allow it to extend past the
rail to a small distance (maybe a 1/2") from the house. Fasten the pickets
onto the rails as normal.

Jon