My go-tos on my Arch laptop are Doom and Quake, there's tons of sourceports, both focused on singleplayer or multi player for both games and their sequels. You just need to buy the games once so you have the main game files to use with the optimized source ports.
Would run and gun games count? If so, I strongly recommend the Neo Geo Metal Slug games. They're all sold emulated on PC (GOG for the no launcher requirement).
And on a more usual sense of "shooter games", Saints Row The Third (GOG and Humble Bundle) and UNLOVED (original version is a mod of Doom II afaik). And just a note, Saints Rows The Third, from some tests I did a while back, seems to have a memory leak issue in character customization screens when running under default Wine.
What issue are you having with Battlefield 3/EA? I just installed it last week I think I did it via Lutris
Good ol counter strike
I played way too much Counter-Strike in college, and I don't regret a minute of it. Then I discovered the "that's so unrealistic bro" genre of shooters like Squad, Day-Z, Tarkov, and Arma, I never looked back.
If you are talking about multiplayer :
Many options !
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