I don't think there's a way define suffering without invoking contextualized pain and an entity that suffers (a self). A self is a complex phenomenon that has a bunch of additional requirements not least of which is the capacity for bridging multiple moments of sensory input that happen in the now.

One can remember that one suffered because one can experience pain in the now and bridge that into a context where the self exists. Selfing is the trick of consciousness, it's not necessarily the natural state of all creatures with sensory equipment or memory - it's an additional trick that only a handful of creatures have.