In conclusion: both the game and the anti-cheat will be able to inspect your Linux system with the same level of access as your current user (for example, both will be able to inspect your $HOME or list the running processes).
If you want to further restrict how the game and the anti-cheat interact with your system, you can run the game inside Flatpak.
For example, if your game is available on Steam, you can use Steam’s Flatpak: https://flathub.org/apps/com.valvesoftware.Steam.
If your game is not on Steam, you can use a launcher such as Heroic: https://flathub.org/apps/com.heroicgameslauncher.hgl.
Keep in mind that anti-cheat systems can detect that you are in a sandbox (for example, within Flatpak) and prevent their own execution, which would make it impossible to play the game, as the anti-cheat system would not be running. I can’t really confirm whether this is the case with, for example, BattlEye or Easy Anti Cheat, although I seem to remember that EAC has no major issues with Flatpak.
However, keep in mind that Flatpak cannot protect you from certain things, such as X11 “security flaws”, so even within Flatpak, the anti-cheat could escape the sandbox if it really wants to. To prevent this, you should modify the permissions of those flatpaks to prevent them from using X11, and then remove access to any non-essential folders.
Please note that a sandbox is not considered inherently secure, and you should never execute potentially malicious code (or code you do not trust) within a sandbox.
In my personal opinion, I believe that by modifying the permissions of, for example, Steam’s Flatpak, the risk is “acceptable” since Flatpak significantly restricts the level of access (thanks to bubblewrap). But really, the only “viable” solution is to use a virtual machine if you are truly concerned that the anti-cheat could access your system.
I hope this information is useful to you, and again, I apologize if at any point you felt that I disrespected you or treated you incorrectly. I’m happy to see new people who want to join the world of Linux, like you!
Please do not hesitate to ask again if you have any further questions on this matter.