A few points that come to mind:
There's nothing that says open source programs have to run on anything even remotely resembling modern desktop PCs. It just means you have fewer people who can do anything with it. Well, until somebody gets curious and ports it.
On the other hand, it's not really open source if only a few people can see it, is it? It sounds like the term your bosses are fishing for is "non-disclosure agreement", which makes your customers legally liable for big trouble if you share with them and they pass it on to the rest of the world (or your competitors). Obviously, the terms of that NDA can't look a lot like the GPL, since much of the GPL is really about redistribution. In short, if you want the whole world to see it, GPL it; if you only want feedback from your customers, NDA it.
The trick to switching to an OSI-compliant license could be whether you can convince your customers that they're paying (presumably) thousands of dollars for a handful of CDs, some printed manuals, and a few support calls. It's a business model that can work for at least the short term (see Red Hat), but I can definitely see it'd be tough to switch mid-stream. And, as you noted, if your customers are competing with each other (which sounds likely), there's a chance that more than one will not appreciate having their "secrets" revealed among others in the trade (even if the only secret is that they've been using the same underlying software all along). It's customers--it has nothing to do with rational thought processes.
I'm not aware of any restriction to licenses (including the GPL) that say that you can't distribute proprietary modules alongside open stuff; you just have to be sure you keep them carefully separated. There is the issue that the API to the proprietary bits (or significant portions of it) may be evident in the other code, so, again, be careful.
Finally, yeah, this smells like AOLism: dropping the S/N ratio in a significant discussion by saying "me too". If that's all they're up to, steer them clear of the term "open source". If opening the source is not what they're doing, it's not what they want to call it, and they'll avoid all the customer relations nightmares mentioned above.
Our scientific power has outrun our spiritual power. We have guided missiles and misguided men. —MLK
$email =~ s/0/o/; # The K5 cabal is out to get you.