2x4' is not bad as daytime-only fencing. You'd want to make real real sure your flock is locked up by dusk every night.
Its main weakness is that things can reach through -- baby chicks can pop out, chickens can poke their heads thru and get them ripped off by a predator they didn't see coming or couldn't get away from b/c not getting their head unstuck from the fence fast enough, and predators can reach arms thru and rip apart chickens who had the misfortune to fall asleep too near the fence.
Thus it is safER if you can add something small-meshed to the lower 2-3'. Even if it is not super strong (e.g. chickenwire) it will at least keep chicks in and slow down the reach-thru problem, which will avoid most incidents.
If you want to add one or more electric wires (on standoff insulators) to the fence you certainly can... you would not want to use electronet for a permanent installation, unless you have 'renter' issues, and you're right, it won't work for you in wintertime.
Remember you will also probably oughta do something to keep critters from digging under the fence. I'd suggest an apron of fencewire maybe 2-3' wide laid on the ground outside the run fence, strongly affixed to the base of the run fence and with the end securely pinned or weighted down. You can cover it with whatever, or leave it for the grass to grow thru, as you wish.
Good luck, have fun,
Pat