Welcome to the Q&A Forum

I think your analysis and recommendations are right on target- don't second guess yourself unless your client is willing to gamble. That said, there are creative ways to get high quality links through networking. The line in the sand between black and white is in constant flux and this is a great example of where this battle continues to rage on. If those inbound links that gave your competitor a boost were industry related, I might think that you've met your match and would need to work harder.  However, the unrelated industry links may not have caught them yet, but are you willing to take that risk?  Is your client? Much of the work in SEO is articulating the opportunities, cost, and risks of chasing the dragon.  If you move too quickly, you may get burned.  Arguing the slow and consistent approach isn't always the best route for a given situation, but arguing against the easy and sly SERP climb is.

Report the link exchange system to Google and they may take action on it.I would not follow suit, you'll put your site at risk if you do black hat SEO. "The websites linking to the competitor are not related sites, or particularly well ranking sites, so shouldn't really give a boost, but it's obvious they are." Just because a site is not related or ranking well does not mean anything, they are still passing PR and they are probably using the correct anchor text.