Just in case someone sees this later, LAX has 9 terminals (1-8 & International, though international flights leave from others too). Terminals 7&8 are at the end of the loop. I didn't want anyone flying United to worry that there were only 6 and they had to go elsewhere.
There's a road work project happening right now, with a warning that shuttles can be delayed up to 30 minutes both ways. Instead of taking the rental car shuttle, you should look for a hotel within walking distance of your rental agency. The hotels on Century are all pretty equal, and with an early flight it sounds like you won't be in the room for long anyway. I wouldn't want to add the extra step of taking the rental car shuttle back to the airport, then taking another shuttle to the hotel when many rental agencies are right next to hotels. I especially wouldn't want to do it with the uncertain traffic delays happening right now.