I would check bags through but be sure to show the bag claim checks to CZ at check in with them, since you must see an agent anyway. (SQ takes note of the numbers at LAX and then verifies that they have the bags, telling me that it's all OK in the lounge. Maybe CZ does it the same at TBIT.) With no worries about getting to baggage claim in LAX before your bags do, you can use the T5 SC until it's time to go to TBIT for check in and their lounge. Leaving the SC (or AS Boardroom, accessible by the airside tunnel to T6) around 6:30 should be good timing.