Flew out of LAX on AC very recently. The Alaska lounge in terminal 6 is open at 5am. It's close to the AC gates. If you have access via Priority Pass or status on Alaska or AA it may be a good option -- not sure $50 for a day pass is worth it. Keep in mind, you'll have to eat food outside the lounge (annoying but easy to keep distant from others).

Regarding hotels, the restrictions are generally against "leisure" travel to red counties (including LA county), so if you have a reason for travel that is different than that, I would suggest contacting the hotels directly since there are exceptions for overnight transit. Most hotels in the area have developed an decent understanding of the regulations but there are some differences of interpretation between hotel chains and individual hotels.

Good luck!