Same problem here, except it was the starter that was replaced.
Have tried this.
http://www.fixya.com/cars/t1239772-1990_le...no_check_engine
..and some key tricks involving opening and closing doors locking and unlocking and generally dancing like a monkey. still nothing....
...I called Scorpion (the company is gone but they still have support in the UK on +44 01257249928) no joy there either, they said hold the black button on the remote for 10 seconds. doesn't work.
next I'm calling Lexus in hope of some inspiration.
Anyone got any pointers/had this problem?