There is rarely evidence of conspiracies. The police decided that rather than admit the truth and then suffer all the fallout that would stem from that, they would instead cover it up. No harm done. A secured premises a mentally ill young woman all pointed to an easy get out. What they did not reckon was the familial issues that would stem from such an action. They did not know the family and their tenacious personalities.
Then there was the question of the inquest. This probably required some elements of re-staging. They had to get their story past the Coroner. T
hey had already lied at the opening saying it was four murders and a suicide to the coroner but were continuing their investigations which was probably required because the family were 'on the case'. In the event the arresting, charging, and conviction of JB negated the inquest.
There would have been a huge amount of work required in relation to the accidental shooting. This would have required many man hours of work and to some extent the public humiliation of EP and their errors.
The easy way out looked at first glance an easy option.