There isn't enough room on my phone to restore - as the process is forcing me to update my software. Is there a way to restore my phone without updating the software?

Hello tjapplesupport,

Welcome to Apple Support Communities. We understand having issues with your iPhone stuck on the Apple logo, and we'd be happy to discuss this situation in more details.

We understand that your iPhone is stuck on the Apple logo, you already tried to force restart it but it didn't help, and when you connect it to your computer you get the option to update or restore.

If you choose to restore, restore erases the information and settings on your iPhone and installs the latest version of iOS software. 

Restore your iPhone, iPad, or iPod to factory settings

"If there isn't enough space to download and install an update, your device temporarily removes some downloadable parts of installed apps. You shouldn't need to remove or relocate any content. If you still need more space than you have on your device, there are a few ways you can update:

Plug your device into your computer to update using your computer.

Delete content from your device and try to update again.

Manage the storage on your device with recommendations."

If you need more space for an update

So if you choose restore or update, the latest software will be installed on your device. Keep in mind that choosing restore will wipe out your data, so if you don't have a backup, you'll not be able to retrieve your information.

We hope this information is useful. Kind regards.