Can't restore iPhone backup because my MacBook is too full...?

I've recently done a hard reset of my iPhone 5 because it needed repairs, but saved a backup within iTunes on my MacBook first. However, after the phone was fixed, when I tried to restore the backup, I received the message: "iTunes could not restore the iPhone because there is not enough free space on this computer to save the backup."

This is illogical: the backup IS on my MacBook; why would it need more space? Is this an error, and is there any way around it?

I can't free up much more space on my MacBook. Currently we have 42GB free of 249GB. I think the backup is around 10GB, for a 16GB iPhone, and of course the phone is currently near empty at factory settings. I've also tried plugging in my portable hard drive during the process, as I've heard having the extra storage attached might help (around 400GB free), but it made no difference.

I have tried to move the backup to another iTunes library on our new, mostly empty MacBook, but iTunes is not recognizing the backup, even though I've stored it in the default location (Library\Application Support\MobileSync\Backup). This could possibly overcome the problem if it IS to do with our other MacBook being too full. iTunes won't give me the option of restoring the backup as I've not technically made a backup of the phone to that computer.

Can anyone think of a work-around, or has anyone else experienced a similar problem? All I really need is my contacts and some of my notes!