Hi Kazi and welcome to the Lexus Owners Club !
The problem your experiencing is a not infrequent one when working with after maket remote starters .As Mason said, I would put a pretty sound bet on the wiring installation of the after market starter.
You say that you talked with Lexus Roadside Assistance, and that their suggestions were working. Can you elaboate?
If you continue to have problems, It is frequently the action of the engine immobilizer. The immobilizer is designed to keep the engine from starting if someone tries to steal the car, the engine will not start. If your in the car and you get out to go into say a store, and you keep the FOB inyour poctet, its possible that when you get in it will die on you.
I have had 3 - ES 350, all with aftermarket remote starts.Allof themhave required some fine tuning. If you keep having problems go to your installer and get the schematic from them. If they won't give it to you, go to you Lexus dealer.My dealer sells both after market remote starters as well as the factory device. I don't recomed the Lexus factory product.
Paul