bzcat

Given your general unfamiliarity with LAX, I would clear TSA in T6 or T7.

T6 and T7 is connected behind security above ground in a clear and obvious way.

If you clear TSA in TBIT (or T4 for that matter), it will involve long walks above and below ground to reach T7, requiring multiple trips up and down escalators/elevators and changing from TBIT arrival up to departure level, back to arrival level at T4, then below arrival level to reach T5 tunnel, and then up to arrival level at T6, and finally up again to departure level to reach T7.

Since you are on LH-UA connection, I assume this is on one ticket and LH will tag your luggage all the way to your final destination. This is what I would do:

1. Collect your luggage at TBIT and clear custom.
2. Re-check your luggage at the re-check station just outside sterile zone but before leaving the secured area.
3. Once outside the secured zone, take elevator or stairs up to departure level and then exit TBIT.
4. Walk outside the upper departure level sidewalk to T7, it will take about 10 minutes at a leisurely pace.
5. Enjoy the sun.
6. As you walk by T6, take a look at the TSA lines. If it is short, go in there and clear TSA. If not, keep going to T7.

You can use the TBIT-T4 connector and T4-5-6 tunnels in the unlikely even of rain.