Best workaround would be to use the (free) shuttle buses going from the terminals to either the Aviation/LAX metro station ('G' bus) or the Lot C Parking/LAX Transit Center on 96th St. and taking Uber/Lyft etc. from there. This will add about 20 minutes to your transit time ( at least, more during heavy traffic).
Alternatively, use any of the airport hotels nearby on Century Blvd as pickup point - the Radisson (I believe it's no longer a Radisson, though but was renamed "Concourse Hotel" last month or so) is even in walking distance - about 350m - from T1.