You can run the Big Fish Games Game Manager on an Ubuntu Linux system under Steam. You can install Steam as a Snap package from the Ubuntu App Center.
Steps:
When I started the install process, the installation appeared to hang at "Execute: C:\Program Files (x86)\bfglient\epoch.exe". When I clicked on the title bar for the window, I saw "About Wine", so it appeared to be using Wine, which I had previously installed on the Ubuntu system.
But I waited and eventually the installation succeeded, though I think it took over 15 minutes. I closed the window and then when I went back to the Steam window, I saw bfginstaller32_s1_l1.exe was still there. This time when I clicked on it, the Big Fish window where I could sign in to Big Fish Games opened fairly quickly.
Note: if the window goes blank or displays just a white background, minimize it by clicking on the "-" in the upper, right-hand corner of the Big Fish window and then switch to it again, which you can do with Alt+Tab — continue to press the Tab key while continuing to hold down the Alt key to cycle between open open windows until you get to the Big Fish window.
Initially, you will see "No games to play," but if you have already purchased games that you've played on another system, you can click on Purchase History and install them on the Linux system. When I installed the Big Fish Games Manager under Steam, I had my wife check one of her games, Aquascapes, and that ran fine. I then closed the Big Fish window and the Steam window and reopened it. Note: if you want to change the name that appears for Big Fish Games in the Uncategorized list, you can right-click on the entry, e.g., bfginstaller32_s1_l1.exe, and choose Properties and then change the "Shortcut" value to something you prefer, such as "Big Fish Games". When I reopened Steam and started Big Fish Games, I saw the Aquascapes game under "My Games".
Other options to use for running Big Fish games under Linux are to use Lutris or Bottles, which are fee and open-source software that can allow you to run games developed for Microsoft Windows on a Linux system. Alternatively, you can use CrossOver, a commercial solution. CodeWeavers. CrossOver is specifically optimized to run Windows software like the Big Fish Manager.