I spent an hour or so tweaking and fiddling with the Combustion this morning. Some progress, but still not sure this is for me.
As mentioned, it sounds a lot different through the amp at volume than it does acoustically on the couch. I was able to call up a variety of tones, some of them usable to me through the amp. The action still seems rather high to stop the rattling, which makes it difficult to play. Curiously, some tones sound different at different parts of the fingerboard. For example, the 8th fret on the E-string sounds dull and lifeless, while the rest of the string is bright.
My technique was partially at fault for some of the issues. Again, someone mentioned to 'let the amp do the heavy work'. I tried this, playing with a lighter touch. It definitely helped.
As I don't really have a requirement for bright, modern tones, it seems that I'm attempting to cure the effect rather than the cause by even using such an animal as the Combustion. I recently passed on a used Super P... mistake!