If you use icloud on your iPhone 4 you should be able to sort your new phone out without using the mac or any computer for that matter.
Sync email, contact calendars etc. to iCloud on iPhone 4. Backup to icloud as well if you like.
As far as I understand you don't need to restore your 6 from the 4 backup. Sign in to icloud on iPhone 6. Everything's there already. For apps, you just gotta sign in to the same App Store account on your 6 as your 4.
I hope this is what you are trying to do, otherwise I missed something in your rant!
P.s. Error message...when does it appear? Not enough space where?
If you use icloud on your iPhone 4 you should be able to sort your new phone out without using the mac or any computer for that matter.
Sync email, contact calendars etc. to iCloud on iPhone 4. Backup to icloud as well if you like.
As far as I understand you don't need to restore your 6 from the 4 backup. Sign in to icloud on iPhone 6. Everything's there already. For apps, you just gotta sign in to the same App Store account on your 6 as your 4.
I hope this is what you are trying to do, otherwise I missed something in your rant!
P.s. Error message...when does it appear? Not enough space where?