Think about it, BB had to think like L to be L.

He had to think, like the man that he accused of his best friend's death, and become him, in his mindset.

Eventually, he probably had a conversation with Naomi about A... just because he needed to get it off his chest, yet he couldn't break character...

He probably worded it something like this.
"He was supposed to be my successor, him and Beyond. Yet... he killed himself. Honestly though, I think of it as I killed him, he snapped under the pressure of being my heir, which was pressure I put on him... honestly I feel horrible about it, I should have let him know I was there for him..."

And in that moment, BB realized all of what he had done, was for nothing... it only added to his brother's hurt.

Yet... he didn't want to turn himself in, nor did he want L to know it was him doing all of this...

Thus, to keep the case unsolvable... he lit himself on fire