I do detailing work on the side, and I have a process that I use on all wash and waxes that gets the bugs off pretty easily. When I am going to be waxing the vehicle, I pretreat some of the areas with full-strength (undiluted) super clean degreaser. I spray front bumper first, the front cover of the mirrors, the rocker panels, wheel wells, etc. I then pressure wash the vehicle saving the front bumper for last. I usually get close to if not all of the bugs off in the pressure washing step. The rest usually come off in the washing step. Mind you, any paint that comes in contact with the degreaser is going to need to be waxed. It seems though that this isn't so much a problem of physically having bugs on the vehicle so much as having an imperfection in the paint from the bugs for whatever reason.
With that being said, I would probably tackle it in this order to see what it takes. I would first try to clay the vehicle, since you have already done that, you might could try a more aggressive clay. I would probably just go ahead and move on to a polishing step though. If you have a DA polisher, I would use a foam polishing pad to see if that takes care of it. If not, polish with a microfiber applicator. I use Meguiar's Ultimate Polish on mine, but it is very mild and may not get off. If that doesn't work, I would move on to Meguiar's Ultimate Compound. Don't worry, this is a very mild compound, and in my experience, it doesn't usually even leave much of a haze. If using a DA polisher, I might try it with the polishing pad first and then move to a cutting pad if that doesn't clear it up. If you're going by hand, go ahead and grab a Terry cloth and get prepared for some serious elbow grease. I have used this stuff by hand. It works well, but it takes a lot of work to get the desired effect. If that still won't remove it, which I highly doubt will be the case, you might need to step up to a slightly more aggressive compound. In doing so, remember the 5 foot rule. On difficult imperfections that require aggressive treatments to fix, don't keep working and working on an imperfection that can't be seen from 5 feet away. You'll end up doing more damage to the clear coat and risk having a portion of your paint look slightly different than the rest. After compounding, remember to follow it back up with either the less aggressive compound and then the polish or just the polish depending on how far you took it.
