I think the invisible fence/collar is your best bet.
Like bud16415 said. A snow fence is for forcing the snow to drift where you want it rather than where you don't. A properly planned and working snow fence is actually a pretty amazing bit of engineering, weather planning and age old wisdom, and very very few people get it right. In my area we have a lot of wind.... and if it snows even just a little the wind blows it until it hits something and then deposits it usually on the south side... that something can be a rock, 2' tall weeds or a 4' fence. If nothing has stopped the wind from blowing the snow it will deposit ALL the snow in that area on the south side of the fence until there is no snow left OR the wind fills it till its flat. Usually resulting in a drift the exact height of the "something that blocks it. So when planning a snow fence your really trying to get the wind to hit something and deposit the snow where you want it rather than, say, your garage door, driveway, etc. It's mostly old Farmers that understand this and you can tell right away who does not when you see somebody pile snow on the North side of their drive way.... there by creating a "Snow Fence" that will fill back in their driveway to the exact height of their piled snow. Resulting in them giving me a call and asking if I can dig them out with my tractor again.... I have one neighbor that refuses to listen to me and has decided since that he just needs a big 6 ft fence on the north side of his drive to stop the snow from drifting. I'm going to send him a bill if it piles up the whole 6ft ;-)
So back to the original post. If it's truly the performance of a "Snow fence" your looking for the only way to have a gate you can use would be to make one that the snow can easily pass through and not create a drift on the other side. However because snow piles on the other side of even a small blade of grass this creates a problem for preventing your dog to get through. Here we make gates out of a single or sometimes double bar swing gate about 2 feet off the ground so that the snow that hits it falls down and then is blown under and past the entry. Then for livestock we put in those "Cattle stoppers" in the ground to prevent them from crossing it. I've never owned a dog so I don't know if the cattle stopper would deter a dog(probably not they're too smart), but those also fill with snow occasionally and can be a bit trouble some without the bar gate I mentioned. Cattle do usually remember where they are even if they covered but young ones forget and have trouble. You might be able to get away with a gate that is made out of wire, but make sure it's fairly large and the higher off the ground the better. Make sure you keep it clear of any weeds/growth. I've seen chicken wire occasionally cause a drift but if anything has grown on it at all it creates a snow fence.
We don't ever get 4' of snow here but every time it snows we easily get 4' drifts. If your not getting that kind of wind then I would forgo the fence all together and like others have said go with the invisible fence.
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