I do not believe that an isolated section of copper plumbing should be considered a 'piping system', though if the inspector is calling 'a pipe, a mixing valve, and a shower head' a piping system, then no rules are being 'made up'. If the inspector is arguing that this short section is a metallic water piping system, then code requires that it should be bonded.
I believe that equipotential bonding around a bathtub, including all metal fixtures, supply pipes, and _drain_ pipes, is a _good idea_. I've seen enough reports of strange tingling in the shower, the EPRI research on 'contact current', etc. to convince me that this is worth doing. In this I agree with dbuckley.
Having said the above, I do not believe that code requires this sort of bonding. In fact, I believe that the requirements of current code, as generally interpreted, actually make equipotential bonding issues _worse_.
Under current code, if you have standard metal piping, you would bond the copper supply pipes but _not_ cast iron drain pipes, which IMHO actually _increases_ hazard at the tub (by having some bonded metal and having separate _non-bonded_ metal in contact with earth.
The focus on the very large conductors for bonding metal water pipes is a historic artifact from the use of common metal underground water supply, and the use of the water pipe as a ground electrode. When you have multiple services in multiple buildings, all sharing the same metal water pipes, then it makes sense to size the bonding conductor to this 'backup neutral' for the fact that it will very likely carry a significant portion of the neutral current. But for an isolated segment of water pipe, such a large conductor simply makes no sense.
IMHO as we change over to plastic piping, drainage, and vent pipes, code will have to evolve where we actually do have 'equipotential bonding' in bathrooms, but the bond back to the panel will be reduced to the 'circuit likely to energize' the bathroom, eg. the bathroom circuit EGC. Such a code change is not currently supported by the evidence, but my gut tells me that we should be looking in this direction.
-Jon