Different discussions have been posted on this topic before. Randomly, the navigation start-up time will be really long and by the time the backup camera actually shows up, you're going forward already. No resolution has been posted by Lexus on this topic yet, but I have noticed the following, which seems to "solve" the problem.

If you put the car in reverse as soon as you start it (as soon as the navigation animation starts) the camera shows up right away and it consistently shows up right away. I am going to call Lexus next week to tell them this, they might be able to update the firmware to fix this permanently.

 

That's good stuff, I've noticed the time being random for the reverse camera too. I'll try interrupting the boot splash screen and see if it works like yours does.