Vtyjkyuk

Use your caves or catacombs to make ice cellars. Their history goes back to 1780 BC.

^{Source: Early Ice Houses)}

Use ice if you can find it, otherwise pack the snow as dense as possible.

See also this question on History SE.

Finally, tell your son to build the snow man inside.

Just try this - even if I'm not sure if this answer fits your "no alchemy" point:
Before building the snowman just mix some sawdust into the snow, the finer the better.
This will effectively prolong the life of the snowman significantly without the need to build any extra buildings or the like.

See https://en.wikipedia.org/wiki/Pykrete for a sample.

P.S.: I would love to post any original sources here, but I can't seem to find any on the run which don't just coat the snowman in sawdust, not mixing the snow beforehand. But nevertheless this works just fine and lets your snowman stand in the open up until the sunny May or June.

You could convert the water deposit of your main city in an ice cellar.

They were insulated with sand, straw, sawdust, but the water cisterns even made it colder. Imagine something like this full of water (this was the medieval cistern for a city with 30,000 people)

1. Find high enough mountain


2. Get snowman above snow line

For the reference, Alps have snow line at around 2,5-3 km..

In the region where I have grown up, they used to harvest snow during winter from nearby mountains and store it in underground rooms insulated with straw.

Therefore:

• Build a large underground room, so that it is protected from the day heath.




• Fill it with snow and straw, add salt for further cooling




• let your son make the snowman, wearing a good woolen coat




• give the snowman a woolen coat, too (yes, wool insulates from thermal exchanges in both directions...)

In 400BCE, Persia had special structures that collected water in the winter, turned it into ice, and kept it cold for the entire summer:

See: http://en.wikipedia.org/wiki/Yakhchāl
And: http://www.ccsenet.org/journal/index.php/ach/article/view/13822/12244

Yakhchāl (Persian: یخچال‎‎ "ice pit"; yakh meaning "ice" and chāl meaning "pit") is an ancient type of evaporative cooler. Above ground, the structure had a domed shape, but had a subterranean storage space; it was often used to store ice, but sometimes was used to store food as well. The subterranean space coupled with the thick heat-resistant construction material insulated the storage space year round. These structures were mainly built and used in Persia. Many that were built hundreds of years ago remain standing.

By 400 BCE, Persian engineers had mastered the technique of using yakhchāls to create ice in the winter and store it in the summer in the desert. In most yakhchāls, the ice is created by itself during the cold seasons of the year; the water is channeled from the qanat (Iranian aqueduct) to the yakhchāl and it freezes upon resting inside the structure.

Put the snow man in a cooling bath with some salt in it.

Salts like ammonium chloride lower the melting point of water-ice solutions: "−10 °C can be achieved with a 1 to 2.5 ratio by weight of calcium chloride hexahydrate to ice.".

That should keep the snow man cool. You would need a natural deposit of that salt (probably a different salt than just sodium chloride). Mining will probably help there, because many million years ago there may have been sea where now is middle of France.

The knowledge of cooling by adding salt crystals to water would probably be something in the realm of an alchemist, but once somebody (a traveling alchemist) told you, the way to maintain a cooling bath would be rather simple (just keep on adding special salt crystals and change the mixture often).

Example setup: https://www.thoughtco.com/create-a-safe-endothermic-chemical-reaction-602207

See also about Frigorific mixtures.

As a bare minimum, you should take a sprayer with water (or just broom/bossom; or even a bucket of water — depending on the snowman's size) and cover the snowman's surface. Of course, it should be relatively cold outside (so that water soon/immediately freezes). This may deform snowman's surface a bit, but probably will make the snowman somewhat more secure against temporary warmings.

Sorry, if the advice is too obvious.

Update: Per Nathan Cooper, it might be just a myth/mispractice/rumor that covering a snowman with a layer of ice would strengthen it against warmness. Thermal conductivity of ice is actually higher and albedo of ice is actually lower. Or this procedure (covering a snowman with a layer of ice) may serve some specific goals (like durability of small details against wind), but not the warmness-protection.

In addition to the cold-room solutions proposed by others, get your son to mix the snow with sawdust as he makes the snowman.

This will drastically increase the snowman's resistance to temperatures and melting by making a mixture known as Pykrete.

As an added benefit it's also much more resistant to physical damage, in case any pesky peasants attempt to destroy it.

Step 1.
Dig a ditch around the snowman. (So the snowman can't run away)

Step 2.
Tie a rope around the ditch and pull until the snowman and ground under the snowman moves.

Step 3.
Pull the snowman and ground under it to the North or South poll.

Step 4.
Take your son there and hope that he won't get lonely with just the snowman to talk to.

Make a bell-shaped glass container, one for each snowman. Then with the aid of a little magic have a dragon suck all the air out of it in order to create a vacuum. A little more magic is needed to seal it. Then, similarly to interstellar space, thanks to the absence of particles that could transmit heat to the snowman, it should take quite long to melt. Keep it in the shadow, you never know.