The 'burn after reading letter' looks like a whole heck of a lot more than 'a little joke' RL said it was. It sounds so bizarre to me, like RL was jealous of GP. No wonder RL didn't want it to get out to the public.
I think RL wrote this to BL when he got home in Gabby's van after he murdered her and told them what had happened. BL probably kept that letter in his backpack and didn't 'burn after reading'. I bet RL was mortified when that was discovered along with BL's totally unbelievable, self serving confession letter with his plea not to blame his parents...they'd 'lost a son and a daughter".
No, the L's didn't 'lose 'either IMO, they knew where BL was when he showed up in Gabby's van and they knew he chose to end his life by suicide. They didn't lose Gabby, their son murdered her. Yet they had the advantage of knowing this and not the gut wrenching, unending grief, worry, and the millions of questions the Petito family had to endure while their daughter lay out there decomposing in the desert.
JMO