TF2 doesn't have any BR modes. KOTH is a standard arena shooter mode, with teams and respawning; "KOTH" only signifies that the objective is to control a static area in the middle of the map.

The closest that TF2 gets to a BR is its "deathmatch" or "elimination" mode, I don't remember the name, in which there is no respawning and the last person standing wins. However, this is very different from a BR; the maps are still tiny, there is no loot to upgrade your character, and there is no "circle", which is a wall of damage that shrinks to force the remaining players to move towards each other rather than staying put.

Usually in TF2, there's teams and respawning.

For something that might be more applicable to your application, check out Tetris 99, which is a battle royale version of Tetris that Nintendo released earlier this year. Not only does it encapsulate an interface that might work better, but it also (as I understand it) thas mechanics to link you to other players in the set so there's a bit of 1-on-1 competition where people affect each other within the larger competition to last the longest.

I can imagine this being pretty fun if, for example you could also perform some task (answer a trivia question, move the mouse through a maze, etc) that might affect others (such as blank their video feed for a few seconds, or flash it white, or apply a weird color filter). The attention and concentration might make people forget to not blink, and the altering off opponents visuals could throw them off as well.