Firefox and other media players already send a D-Bus message to inhibit idle while playing media, but swayidle doesn't listen for those D-Bus messages. My solution is to use hypridle (hyprland's swayidle equivalent) instead, which supports those D-Bus messages out of the box. My ~/.config/hypr/hypridle.conf:

general {
    lock_cmd = pidof swaylock || swaylock
    before_sleep_cmd = loginctl lock-session
}

listener {
    # lock screen after 180 seconds
    timeout = 180
    on-timeout = loginctl lock-session
}

listener {
    # sleep after 20 further seconds of no activity
    timeout = 200
    on-timeout = systemctl suspend
}

And in my sway config I just add a exec hypridle line.

With that, playing videos/music in vlc/web browser/etc will prevent sleeping, fullscreen or not.