I use awesome, and there's a way you can assign particular programs to specific tags, so that they always open on that tag. I'd be surprised if i3 didn't have a way to do the same, given that even dwm can do it ;)
Then, there's a third option in addition to the two you mentioned: spend two minutes (if that) doing the one-time configuration of having the wm reserve a tag for your browser (and perhaps other GUI apps that you anticipate frequently needing/wanting to run full-screen). In awesome, you'd do that by adding an entry to your global rules table like this one (in rc.lua):
{ rule = {
instance = "firefox"
},
properties = {
screen = "HDMI1",
tag = "3"
}
}
Which will configure awesome so that any time a new client becomes managed by the wm, it will ensure that its window opens on the XRandR screen "HDMI1", and specifically on that screen's tag "3". ("3" is a string here because, while tags are by default "integers", "1" - "9", they can be named by any string afaik, and many people will rename them things like "www", "irc", "term", etc.).
You can also use awesome-client(1) to send Lua code to a running awesome instance for execution within the wm's context. This could be tossed into a shell script (and parameterized as you see fit) for convenience, if desired:
$ awesome-client <<EOF
require("awful").spawn("firefox", {
screen = mouse.screen,
tag = "3",
switchtotag = true
})
EOF
The same awful.spawn() call you'd send to awesome-client can also be bound to a keystroke or a menu entry for quick access.
As for me, I tend to only use tmux under X11 if I'm worried about losing an ssh connection and therefore (without tmux, of course) the ssh session. However, I have DEC vt520 glass TTY that connects via RS-232 serial cable to a couple of servers (one OpenBSD, the other FreeBSD) and, while the terminal is pretty damn featureful, it has limited scrollback and awkward copy/paste functionality. So, on that terminal I use tmux quite a bit (although it seems to be rather chatty and during periods of heavy pty traffic can sometimes wind up spitting a garbage glyph on the screen here and there, but I think I may just need to increase the baud rate (despite already being set to ~56 Kbaud! cha-tty!)). The vt520 also only supports 2 sessions at once directly (and I can't even figure out how to get the OS to recognize the second one -_-), so tmux's multiplexing capability is also greatly appreciated when I'm basking in the amber glow.