If I understand correctly, you have an FP1 first edition, and you haven’t applied the partition layout upgrade. If you’re unsure about whether you’ve updated the partition layout, check under Settings > Storage. In the new layout, the total space in internal storage will be a single 13.37 Gb bar with different coloured blocks. In the old layout there will be internal storage (1Gb) and phone storage.
In the old partition layout, there would be a section of 1 Gb reserved to install apps. Even though there may be more free space elsewhere on the device, this can’t always be used to install apps. The new layout works around this by merging different sections of storage.
On the page you linked to, there is another link to the instructions on how to upgrade the partition layout. It’s the link just above the warning. However, the process does perform a hard reset, which you mention you want to avoid.
To work around the problem you can try tips included on this page: How are apps stored on my Fairphone 1 (Internal, Phone, and SD Storage)?. Most useful is removing apps you no longer use. Moving apps to the Phone storage can also help, though not all apps can be moved.