An addition to the record, as Google returns this issue as one of the top results for "python install \"Error 0x80070005: Failed to register bundle.\"":

In my case, I had removed access for all users on the HKCU/RunOnce key and was getting this error.  The purpose of removing access to that key is so Windows doesn't automatically relaunch all running programs on restart, as the setting to disable it does not exist in some earlier windows 10 patches.

HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\RunOnce

After restoring permission to my user, the package installed successfully. 
 During installation, it was observed that the path to a temporary copy of the installer executable was added as a string value under the RunOnce key, purpose unknown.  This string value was removed by the time installation was completed.

While I disallowed access to this key manually, I suspect that some anti* software may also block access.

Binary:
python-3.7.1-amd64.exe
SHA256 86E38B5F819F4141A7FA14DF39AB33E2AA97074E904A48AD498519AD3E69D017