There's nothing to stop you clearing security at T4 and then using the connector to walk to TBIT to access the lounge there. In many ways it's a lot quicker, especially if you have PreCheck, and also it avoids awkward questions from TSA who sometimes are reluctant to allow you to clear security in TBIT if your flight departs from another terminal.

You can also clear security in T6 of course. I have done that a couple of times when traffic at LAX has been hellish and it was quicker to detour via the shortcut to T6 / T7 than to crawl past each terminal sequentially but at a snail's pace. If you do clear at T6 then the quickest way back is usually via the tunnel.