I've not stayed at the Hilton LAX yet but ...

I'd return the night before to the Hilton LAX HLE to avoid the car parking charges at the hotel.

Additionally, you only have one variable to deal with - the hotel shuttle - and not the check-in line at the airport, additional traffic and so on. I don't know but expect that the hotel shuttle will be more direct and therefore quicker for your flight connection.

The only issue is that the HLE may not be able to accept a return late enough for your return to the hotel. If you're planning to check in early and stay in on your last night, there's no issue.