I suppose one could coat them in clear resin, to eliminate the sublimation. They still have to be kept cold, though, or you get a snowman-shaped snow globe. (hmmm…)
I think the best is doing snowman taxidermy, so to speak. Use the resin, but make a mold instead, and cast copies using some sort of aerogel. Then recycle the snow. It would capture the ephemeral-ness, but make them last indefinitely, unless you touched it too hard.
Why not instead use the tried-and-true lost wax method, except instead of wax, you’re using snow.
I actually thought about that, but I wasn’t sure if there is a resin which is suitable for the application, as it’d need to be below 32F (preferably more like 0F, more or less) at application as a liquid – otherwise, it’d melt the subject. 
Use 3D scanning technology and print replicas using a 3D printer and white resin. Have to paint the eyes nose and arms, but doable. Bonus: you can scale them and sell them on ebay to model railroad and diorama builders!
I once made a snow Dalek to hold up a Valentine for my wife.
For the museum:
Three days later. “The Heepada” as I named it is denuded, blind, stripped of all accoutrements. Returned to its original snowplowed existence. City snowmen have it rough.
Someone can still use a can of black spray paint to paint a snowman on the side (see my earlier post)
Lacking means to preserve, but I womder if anyone has kept a snowman alive by adding more snow as the old snow melts.
Let’s say you start an amusement park called Frostyland. You want a real snowman at the front gate to welcome the guests. Every night after it closes, you pack new snow around it and rebuild it as best you can. You have an indoor freezer that can create snow from water even after the weather gets warm. Is it still the same Frosty after all the original snow has vanished? That’s a question for the metaphysicians.
The Snowman of Theseus.