1) Tolls are regressive. I believe cities in the US have power over some taxation so why not offset the extra cost by reducing another tax? e.g. property taxes in low value areas could be cut, or sales taxes. Alternatively spend all the toll revenue on heavily subsidising public transport so that it is cheap and plentiful.
2) Other cities have had tolls and it did not effect the traffic. Why not just raise the toll until it does effect the traffic? In London the charge for breaking Ultra-Low Emission Zone requirements will be £130/$160 after 2020. Raise the toll to $100 and it will have an effect.
Sounds though that the underlying issue is that LA's urban geography doesn't lend itself to public transport so that whatever you do people will still have to use cars if they want to travel any reasonable distance. Short of undertaking a Shanghai-metro level of construction (from nothing to the world's largest rapid transit system in 20 years.)