It's worth showing your incoming boarding pass at the AA/QF lounge in T4. They might just comp you. If they don't and you still want to get in you can buy a 'One Day' Pass for $50US. Pricey, but if you can't get a shower elsewhere before your onward flight, it can seem like a bargain.

If you have priority pass...the closest appears to be Terminal 5 (probably the Delta lounge which *also* has nice shower rooms but is otherwise unremarkable).

Does anyone know if the underground tunnel between T4 and T5 at LAX is still open?

I'd wager you would have to go through security at T4 (since your boarding pass shows a flight leaving from there). If the tunnel is open, you take that to T5 (you remain in a 'secure' area the whole time then) and then try the lounge there. The Delta lounge is in the middle of the terminal near some shops (right hand side, small elevator or stairs to 2nd floor lounge...very small Crown Room Club sign).