Fix BadResponseType test for TestLogSearchController

[michellemcdaniel]

• [ ] This issue is blocking
• [ ] This issue is causing unreasonable pain

In https://dev.azure.com/dnceng/internal/_git/dotnet-helix-service/pullrequest/24688, we had to remove this test because it has hardcoded dates in it that were more than 2 weeks ago. Because of how the controller is written, instead of getting the expected exception, we got an out of date error, see failure log: https://dev.azure.com/dnceng/internal/_build/results?buildId=1926171&view=ms.vss-test-web.build-test-results-tab&runId=49828534&resultId=100346&paneView=debug.

This test needs to be rewritten to not hardcode the dates so that the test can be run.

@ulisesh