My guess is that taking out Ingram in a quieter way would have lead to further investigation from the authorities, and therefore more loose ends to tie up. A fake suicide may have worked, but maybe they thought there were too many people who wouldn't believe he was suicidal.

They dug her up for her cochlear implant - they extrapolated the location data and found the places she had most commonly been - the precinct and the subway, which is how they found everyone so fast.