Are there any iPhone cases compatible with full-body iPhone films (i.e. films on the sides)?
Most people I know of that appreciate the invisible shield type skins go for a sleeve as their secondary protection method.

I have seen very nice ones made DIY style from all manner of foam - the more risky a sport, the more padding you can include. It's cheap, light and only as bulky as you need. A simple strap will keep your phone in the sleeve and you can waterproof with a bag as needed.

The nature of the soft sticky films is that they grip most hard cases and result in the damage you are seeing to the skin when a hard plastic case gets fitted over the phone.

If foam isn't your preference, sleeves from felt and leather are also widely available from reputable retail and internet sellers, etsy, ebay. Don't overlook the potential to have your local craftsmen or craftswomen make a sleeve to your exact specifications for a reasonable cost.