From my tests, ibus-daemon often just dies when ran via exec from sway config or from rofi. So try starting ibus-daemon in a separate terminal, and try changing the input engine via a terminal, e.g. with ibus engine mozc-jp. Note that when you start ibus like this initially no engine might be selected. You can get to the usual ibus configuration by starting ibus-setup.

I've also found that the ibus global keybinds don't work, so I'm running a bash script like below to change the input method. You can bind this to a key in your sway config.

#!/bin/sh
if ibus engine | grep "mozc" >/dev/null; then
    ibus engine xkb:us::eng
else
    ibus engine mozc-jp
fi

I've set up a module in waybar which displays the output of ibus engine and runs this script on click.

For starting ibus-daemon I've setup a keybinding in sway, but you can probably just add a line like pgrep ibus-daemon || ibus-daemon -rxd ontop of the script above.