For a different point of view, I would stay at one of the hotels just past the entrance to LAX. Then you can walk directly to the terminals. I only had 1 rollerboard with me and I walked from the 4 Points Sheraton in about 20-25 minutes. If Friday morning is as congested as mentioned above, you may sit in the shuttle bus for a long time. One time that I used the shuttle, I alighted at the 1st stop and walked over to T4.
As for the lounges, try the OW lounge in the Bradley Intl terminal. AA's T4 and Bradley are connected airside. With a CX flight around 8 am, it should be open for your use as a non-AA Sapphire elite member.