What is the origin of “cold weapon” in the sense of non-firearm?

It seems you're quite right about the translation origin, from Russian. It's a rare term, but which get regularly used by people familiar with Chinese or Russian Military History.

The earliest use in this sense that could be found so far is from

-1862-

Dickens, Charles. All the Year Round. London: Chapman & Hall, 1862, volume 8.

This is a journal partially authored and fully edited by the famous victorian writer.

In a section called "How to make soldiers"

...to accustom him to see clearly through a mass of crossed swords and bayonets and to remain cool while swords flash under shocks unexpected blows and rapid orders. A fencing room in which twenty or thirty couples of men are practising is a fair illustration of a melee with cold weapons

First use I found which also explain the origin of the term in pseudo phonetic transposition:

The cold weapon "kholodnoye orudiye" as opposed to firearms " agniovoye orudiye"

-1876-

The Russian CampaignAgainst Khiva in 1873, Part 1 by Hugo Stumm, Foreign Department Press

It seems to be the earliest reference in English. Anything before that relates to the actual coldness of steel.

Some other references of usage of the term across the last 150 years, still influenced by the Russian origin though.

-1890-

Gustavus Adolphus: A History of the Art of War from Its Revival ..., Volume 2 Par Theodore Ayrault Dodge, page 571

disappeared and only helmet and breastplate remained to the heavy trooper Pistols carbines and musketoons were the firearms of the cavalry a sword or sabre the cold weapon Dragoons carried the infantry musket with a bayonet and came more and more into favor

-1914 -

Infantry Journal - Volume 10 - Page 298

"penetrated and animated by this truth, that in fire the bullet should be cold— and the cold weapon, warm; more clearly stated, that we must fire with calmness and deliver the assault with fury. To sum up, both Russians and Japanese appear ..."

-1982-

Daily Report: People's Republic of China, Numéros 148 à 155 Couverture United States. Foreign Broadcast Information Service National Technical Information Service,

-2016-

Krav Maga Professional Tactics: The Contact Combat System of the Israeli Martial Arts, David Kahn, YMAA Publication Center, Inc

As a bonus if you want to try searching it in cyrilic it seems to be "холодное орудие" (I don't speak Russian)