iPhone backup too big after firmware update

I'm using an iPhone 6s+ with 128 GB internal storage. It was running on iOS 12.4.1 up until recently when I've decided to update iOS to the suggested 14.1.7. Prior to updating, I've created a backup which is about 116 GB in size (yes, my storage was pretty full).

Now here's the problem: after updating iOS, I can't restore my backup onto the phone because iTunes says there is not enough space on the phone.

What I've tried:

• trying out different softwares to delete chunks of data ouf of the backup so it becomes smaller in size: none of them worked.
• downgrading iOS: doesn't work because the old iOS is no longer signed.
• reverting to factory new state: this also didn't downgrade the iOS it seems. After immediately reverting, the backup actually started loading back onto the phone, only to abort at 99% and returning the same error about having not enough space.

How can I now get my backup (or parts of it) back onto the phone if the new firmware takes up more space and I can't downgrade iOS or "edit" the backup?

If you are absolutely sure that you have backed up everything you need on the phone, what you may want to try is an erase and restore:

Settings > General > Reset > Erase all content and settings

What happens is there is some amount of cruft that can build up over time and without going into the whole "buy a movie/show that will almost fill up your remaining storage" so that iOS will then delete the cruft (usually temporary files, VM and OS & app related) files allowing you to then do what you need.

I find it simpler to erase and restore. You will (re)start with a blank slate and be able to restore your backup.

Note that iPhone/iPad backups don't backup the apps, they just keep track of them and re-download them once the data is restored. Also be sure and give your backup a password, otherwise the passwords saved on your iPhone will not be saved.