I have found a clay bar to be poor at removing large contaminates, frequently completely missing large bug pieces.
Clean with Dawn dish soap, wash the effected areas twice. Doing so should remove all wax/detail spray. If not 10% isoproprol alcohol will.
Scrub area with a bug sponge or a chemical bug remover. This should completely remove the bug remains. Rewax as needed.
The bits of bug remaining shouldn't be a problem, its really the bug insides that do the damage. But best to get them off of there. The front of my van as destroyed by not removing bugs in a timely manner. I couldn't compound it enough to repair the paint, and if I tried I probably wouldn't have any paint left.