Good Day. A warm welcome to the HP community. I reviewed the case regarding connectivity issues with your HP sprocket. Superb description, brilliant troubleshooting and terrific observations made before posting. Kudos to you for that. 🙂
To assist you with accurate information I would require more information regarding this:
Did you try to uninstall and reinstall the HP sprocket app and pair it again from iTunes?
Does the iPhone work with other Bluetooth devices?
Does the sprocket connect to another iOS device such as an iPad?
For now, try these steps:
First, we need to eliminate hardware issues with the Sprocket.
Perform a factory restore on the HP sprocket.
Follow steps from this link: https://hp.care/2BH91Go (HP Sprocket Photo Printers - Frequently Asked Questions (FAQs)
Perform the steps from this section “How do I reset my printer?”
Then perform all the relevant steps from this section “Why am I unable to connect my Apple iOS device?”
Perform those steps that have not been done by you yet.
Now check for issue resolution.
If it continues, try pairing the HP Sprocket device with another iOS device like an iPad and check if it pairs and works correctly.
Uninstall and reinstall the latest version of the HP Sprocket app and check if it corrects the issue.
If the issue continues, the sprocket may be faulty and might need to be replaced by contacting HP phone support. I am being honest about it by keeping tour best interest in mind without beating around the bush.