From my perspective as a project manager, the answer to “How do I encourage adoption of (x) development practice” is almost always: DO IT SLOWLY.
Implementing a new development practice can be very hard, especially if it happens via coercion. It's easier if you lower expectations of immediate impact and instead look for opportunities for incremental change.
Since you know how to develop software, it might be helpful to think of Development Methodology Improvement as a project (meta-project?) of its own. The customer is the management team, and the "shipping product" is an improved process. The specific techniques you use to make that product can be as varied as the many patterns you use in a software product. In your case, you're pretty sure you want to use the TDD pattern; but ultimately you should recognize that another pattern could be a better approach.
First, work with the customer to identify the user stories**. (Don't get sloppy here—“fewer bugs” is not specific enough to measure.) Then prioritize them. Then figure out how to show progress in a single iteration. For example, you could show progress by writing tests for a single feature, with a single management representative "pairing" with you, i.e. looking over your shoulder as you write it. This doesn't take a ton of extra time or effort on anyone's part, but the end result is that one feature is more reliable and one manager has more understanding of TDD.
Over successive iterations, you add more tests or involve more coders/managers, or whatever you can do to show continual progress. The key here is that the changes aren't too disruptive to the existing process. The team is still focused on the software, not the process, so management doesn't freak out about wasting resources.
**Don't call them “user stories” if you aren't using that terminology already. Use whatever words you currently use to specify problems that a software product will address.