Expand the Default Web Site in the Connections pane (left side of screen), right-click the application and select Manage Application > Advanced Settings. The application pool used for the application will be listed at the top of the resulting screen. Do this for both LIVE and TEST and make a note of the application pool used for each. I suspect that LIVE will be running under a domain account while TEST will be ApplicationPoolIdentity.
Then go to Application Pools in the Connections pane and take a look at the identity. If they’re different then match TEST to LIVE. You can change the identity by right-clicking the application pool, selecting Advanced Settings, clicking the in the Identity field, click the […] button that shows up, click Set, and specify the user/password. Make sure to stop/start the application pool (Actions pane, right-hand side) after changing the user to restart the application pool under the new context. Wait a couple of seconds and then click the Refresh button at the top right (all the way up near the close button) of the screen and verify that the application pool remains started.
I agree with @Olga, the 404 is strange because we can view the report directories in a browser. What’s listed in the web.config for the application? You can check by right-clicking the TEST application in IIS, selecting Explore, and opening the web.config file with your favorite text editor. The values of interest are SSRSBaseURL and SSRSRootFolder.