Phrase for extreme cold using an animal

Something that might fit along your lines are variations on the phrase X dog night, where X is some number. Thesaurus.com offers synonyms for two-dog night, but strangely no actual definition. The synonym subcategories are cold, freezing, and shivery.

Urban Dictionary lists three-dog nights as:

In the old days of no central heating, a night so cold it took 3 dogs sleeping with you to keep you warm enough.
Example: Man, last night was real cold, definitely a three dog night.

And the website iheartdogs attempts to give more details behind 3 dog night:

The geographic source of this phrase has been debated time and again. No one is sure whether it originated in the Australian outback or the northern reaches of North America with the Eskimos. The meaning, however, is quite clear. The phrase is a rudimentary nightly temperature gauge. Dogs huddled with humans at night for the warmth. On really cold nights, three dogs were called into the bed to keep the owner from freezing to death. The phrase was cemented in literature by Jane Resh Thomas’ book Courage at Indian Deep

I first heard the term on weather forecasts from Gary England in Oklahoma, but have heard it quite a few times.

You have brass monkey weather which is an informal BrE expression meaning:

extremely cold weather.

(Cambridge Dictionary)

From World Wide Words:

• The full expansion of the phrase is cold enough to freeze the balls off a brass monkey and is common throughout the English-speaking world, though much better known now in Australia and New Zealand than elsewhere.
• This is perhaps surprising, since we know it was first recorded in the USA, in the 1850s. It is often reduced to the elliptical form that you give (perhaps in deference to polite society — for the same reason, it has been modified in the US into freeze the tail off a brass monkey).