How to install iOS 13 on iPhone 6s?
This is a risky proposition, but it’s what I would do if I was in your position. Obviously I cannot test it myself. Use at your own risk, etc.


This is a highly important step!!!  Make a backup of your iPhone, either to iTunes or to iCloud. (Ideally, both.) If iTunes, be sure to set an encryption password.
Reset the device: Settings > General > Reset > Erase All Content and Settings.
When it restarts, don’t bother configuring anything, just skip through the setup steps as quickly as possible.
Run the update.
When the update is complete, run Settings > General > Reset > Erase All Content and Settings again.
Restore iPhone from backup.


That should give you your current setup but with the new version of iOS.

Unfortunately, some things do not carry over after a restore, so you will have to reconfigure some apps, but I think it’s the best chance that you have.

If you want to try a less-drastic option…

I would recommend going to Settings > General > iPhone Storage and looking at which apps are taking up the most space. Deleting them manually might help.