I believe Catalyst forbids/ignores live app config changes (sidenote: this has nothing to with model data, just $ctx->config stuff) after ->setup is run. There are probably some hacky re->import(…)/unload things you could try but I don’t think it’s a good idea.
So, for “what goes here?” the easy and clear answer is a new test file with something like local $ENV{CATALYST_CONFIG_LOCAL_SUFFIX} = "other_test" at the top matching your new test config. If you use a PSGI test runner or something instead of the built-in test lib you could probably do this more easily in one test file, in blocks or subtests, but it doesn’t seem like it’s as transparent or a big gain over two test files.