Could the use of anti-perspirant give benefit in extreme cold climate
where sweating can be a significant problem.
TL;DR answer: Unlikely. The issue is the sheer amount of water your body will secrete during physical exercise. It would be impossible for anti persperant to prevent this amount of moisture.
To clarify Anti-perspirants work by:
Aluminium-based complexes react with the electrolytes in the sweat to
form a gel plug in the duct of the sweat gland. The plugs prevent the
gland from excreting liquid and are removed over time by the natural
sloughing of the skin. The metal salts work in another way to prevent
sweat from reaching the surface of the skin: the aluminium salts
interact with the keratin fibrils in the sweat ducts and form a
physical plug that prevents sweat from reaching the skin’s surface.
Aluminium salts also have a slight astringent effect on the pores;
causing them to contract, further preventing sweat from reaching the
surface of the skin. The blockage of a large number of sweat
glands reduces the amount of sweat produced in the underarms, though
this may vary from person to person.
As you say, sweating is an important part of your body's thermoregulation. As you exercise, your body heats up. Your body regulates this heat by sweating. Even in cold climates you will need to sweat to regulate your body temperature. An important part of this is the waters potential to evaporate off the skin (taking the heat with it). If the water didn't evaporate (because it was being stopped by some kind of super antiperspirant) and your temperature didn't lower your body would simply produce more sweat.
The best way to deal with this is to allow this process to work as it naturally should, i.e. to allow the water (sweat) to evaporate away from the skin and though your layers (breathable clothing). This will cool you when you need it to but prevent the moisture from becoming trapped and "chilling" you.