I may have spoken too soon. It seems that the addition did indeed fix the "Not available for testing" issue and I was able to download it and test in via Test Flight for internal testing. However, when I try to submit it for review for external testing I get the following error: This build is using a beta version of Xcode and can’t be submitted. Make sure you’re using the latest version of Xcode or the latest seed release found on the releases tab in News and Updates.

Thank you for your quick reply. Adding the com.apple.application-identifier (and com.apple.developer.team-identifier for good measure) to the entitlements made the game work in Test Flight. I've also submitted a bug report about these entitlements not being documented as you suggested.