a situation that hardly any developer ever finds himself in"To a worm in horseradish, the world is horseradish."
(Sorry, have actual work to do today, so I'm going to reply mainly in epigrams where I can. ;)
pass myself off as... a business problem solver
Nobody is suggesting that. You don't pretend to solve problems for your business. At least, the business doesn't think so. They pay you. They don't pay people just for fun. They obviously think you're solving some kind of problem, right now.
Even within the seventh layer of a nine-layer bureaucracy, your peers have a problem that they are paying you to solve. Figure out what it is and learn to talk about that.
This needn't be self-aggrandizement, either. "I am one of ten interchangeable members of the QA team that keeps our customers happily renewing their subscriptions by finding bugs before they even see them" is fine. You needn't claim to walk on water all by yourself. People like loyal team players.
If you find that you can't talk about your problem without feeling dirty: That's a data point. If part of the job is "follow orders without thinking like a good little interchangeable part", that's a data point. If you discover that the problem you're solving is "win our division's internal war against the Other Division, customers be damned", you have a data point. If you come to the conclusion that you are in fact paid to cause problems, but the company can't figure that out because the intervening tangle of bureaucracy is disguising the fact, that's a data point. Try to keep doing your job well, but as the data points accumulate, and depending on your personality, you may find that other jobs are calling.