Hello, Software Wulf!
I can't say for sure why the Windows version doesn't work with Proton (as there could be some bits of code where Proton doesn't play well). However, you can use the Linux version of the Cook Tool Deluxe and create a native Linux version of the game (assuming that any dependencies for the game are also compatible with Linux).
If I recall correctly, the SteamOS has a desktop mode, so you could set up the tool on the Deck. Alternatively, using the Windows Subsystem for Linux would also set up a Linux environment so you can use the Linux version).