Sounds like you've got the same problem I was so rudely greeted by the first time I unlocked the car after purchasing it. The little plastic sensor on the back of the lock cylinder gets detached or just goes bad, so it doesn't send the signal to the alarm that the door has been unlocked properly with the key.
Post #13 details the fix. I added info in post #16 for how to go about getting at it. It's easily the most difficult thing I've attempted to do on this car, from the standpoint of nearly-impossible access, so be prepared for a challenge!
It sounds like you may need to source spare connectors for both doors if the pass. door is doing it now, too...
BTW, putting the key in the ignition silences the alarm but activates the kill switch. I'm stumped by Lexus' reasoning on that. I'm unsure why your alarm comes back on when you reconnect the battery, though. That leaves me somewhat less than 100% confident your problem is the same as mine was.