How to download non-DRM version of game from Steam to run in WINE?

No, you cannot.

Steam is a DRM enforcing client. It doesn't provide a version of software that is not DRM restricted.

However, you can install Steam on WINE itself. It will then detect it is on a "Windows platform" and install the game.

For some games you can just copy files, but some require Steam for at least some functionality (friend list, server list etc.)

Cogmind uses SteamCloud to backup your saved games and for global High Scores list.

Also, Steam detects WINE as WINE, not as Windows. They specifically made it to also count users on MacOS/Linux who use WINE.

You should run Steam on WINE instead. That way there are less things that can go wrong, and better chances that everything works as intended.