This is pretty much how I feel like it happened, too. I don't think he wanted anyone to know what he had done. When GP's parents started contacting them, I think his story began to fall apart and the Ls, like you said, maybe started asking questions. Or, BL could have blocked their numbers from his parent's phones, or gave them some explanation. He had obviously made other attempts to hide the truth. With LE seizing the van on Sept. 11, though, there was no doubt GP's family, and LE, were taking GP's disappearance seriously. So yeah, the Ls must have been worried what their son had done. They talked to their lawyer friend because they didn't know how to handle things, and the lawyer said to keep quiet, so they did. Even at that point, maybe nobody but BL knew GP was dead, although they were likely getting suspicious. Then came the Sept. 13 press conference with GP's family. Clearly, this was not going away, everyone was putting two and two together, so BL took off.
Now, the question remains: why did it take the Ls so long to report him missing? According to the FBI's final statement, that happened on Sept. 17 during a welfare check. Okay, that makes the Ls look bad. I get it. But we also have SB saying he told an FBI agent that Brian hadn't come home already on the night of the 13th. People can hate on SB, but he made that statement publicly. A majorly risky move for himself and his clients if it was a blatant lie. Maybe the FBI didn't add that date in their final statement because it was never on record. Or, for those who don't trust LE, maybe the FBI was saving face for not doing something with the information sooner.
Either way, if the Ls thought the FBI already knew, then that might account for why they didn't report it again until NPPD came knocking on the 17th. Since taking off and hiking for a few days was reportedly not out-of-character for BL, maybe the Ls didn't necessarily think he would seriously go kill himself, especially if they didn't know he had a gun yet.
I'm not supporting BL or his parents for known or suspected unethical, immoral, or illegal activities. I don't know why it always has to be an all or nothing perspective. In the end, maybe the Ls were just horrible, horrible people. Idk. But the devil is in the details, which we simply don't have.