It theoretically could, as the Linux version of these anticheats run in userspace. We really can’t say for sure, only someone with knowledge of the internals of EAC/BattleEye could tell us if the Linux anticheats are less capable.
The Finals is so comically easy to cheat in that there would be no point for someone to switch to Linux just to cheat. Especially true when you take into consideration the facts that there aren’t many (if any) cheats for this game on Linux, and most cheat developers only have experience developing for Windows. These devs have years of knowledge on hiding cheats in Windows, and none for Linux.