Ultimately I have advised the phones owner to take it to Apple. Apple had previously wanted to wipe the device because it was bootlooping. Now that I have the phone at the restore screen I think they will just replace the home button and screen.
I didn’t end up trying the bluetooth work around as I didn’t want apple to refuse to work on it and soldering stuff to the device was sure to irk them. The bluetooth home button work around could still be a potential solution if someone else runs into this problem. I am skeptical about whether to phone is going to treat the “return to home” function the same way it treats a home button press from a working home button, at least at that stage of setting up the phone.
Thank you to everyone who responded.