The only way I found to install an old version of TestFlight onto my iPhone 4s (iOS 9.3.6) was to backup another iPhone 4s (iPhone #1) that already had TestFlight and then to restore the backup on the iPhone 4s (iPhone #2) that did not have TestFlight. After the restore was complete, the old version of TestFlight restored itself on iPhone #2. TestFlight still works perfectly on both iPhone 4s devices.

We have iOS 5s test units (iOS 12.5.5) running TestFlight. Up until recently, we were able to restore the iPhone 5s test units from a backup to prepare a clean image before the start of new testing. After restoring the iOS 5s units, TestFlight would download and be fully operational. We just restored one of our iOS 5s test units from backup and TestFlight will not reload (Error "Unable to Download Item. Please try again later."). TestFlight still runs perfectly on the iOS 5s test units that we have not yet restored from backup, so blocking the download of TestFlight to restored iPhones that already have TestFlight installed is just an unnecessary/arbitrary obstacle that Apple is creating. On a side note, we also have iPhone 4s units running TestFlight. Up until recently, we were able to restore the iPhone 4s devices from backup and TestFlight would become fully operational after the full restore. TestFlight still runs perfectly fine on the iPhone 4s devices that we have not restored from backup. Why block the full restoration of older iPhones that are still perfectly good and required for software testing?