I'm replying to my own thread, because I've figured out the issue. In case somebody has this same problem in the future, maybe they can use this information.
Though the module was developed using System.Web.Extensions Version=3.5.0.0, and the test site was running mostly 3.5 Framework, the target framework for the production site was 2.0. When I moved the module, which requested System.Web.Extensions Version=3.5.0.0 as a dependency, the target framework for the production site was moved up to 3.5.
I cut the System.Web.Extensions Version=3.5.0.0 inherit call from the module first. Then I switched the target back to 2.0 for the production site. Finally, in the property pages, I removed any references to assemblies at version 3.5. Once these tasks were accomplished, I ran (without debugging) and it fired up just fine, with the module working to boot, as soon as I'd revised the Module Definition Control.
I feel like it's something I should have spotted right away, but... oh well. Hopefully my mistake will help someone else in the future.
- D