Iphone 5 Died, trying to restore to my old iphone 4

Both my 5 and 4 used to show up in itunes. I backed up my 5 the night before it died. I thought it would be easy to install the 5 back up onto the 4, but alas, I haven't been able to do it.

I was using the 4 a mini ipad if you will. I just used it on wifi and all was well, it was still running some version of ios 5.

When I hooked up the 4 to itunes in hopes of installing the back up from the 5, the restore menu in Itunes only gave me the option of restoring the old back ups from the 4, not the new back up from the 5. So I thought I'd force the 4 into recover mode, recover it,and hope itunes would then show the option to use the back up from the 5.

I did that, but I guess the servers at Apple are so overwhelmed updating everyone to ios 8, the progress bar said it would take 25 hours to download. This is with a 40 MPS connection (I checked), so I'm sure it isn't my internet connection.

So I took the 4 to AT&T and just had the SIM card activated and I'm using the phone. I'd still like to get the 5 backup onto the phone as I'm missing a lot of stuff on the 4 that was on the 5.

I guess the 4 had previously downloaded the update to 7.1.something so I let that run last night. I called AT&T and talked to Apple who said I can retore at the store and I won't be subject to the 30 hour download. I have an appt tomorrow. BUT...

Now the iphone 5 does not show up as ever having existed in itunes. I don't want to come home with the restored phone and not still not be able to load up the back up. I was thinking maybe I should try to copy the back up to my laptop, take the laptop with the back up and the 4 to the genius bar so we can load it up there.

But how do I do that? How do I transfer the back up file? Since the 4 is not "married" to the laptop itunes, will it work? How can I get the iphone 5 back up to show up in either the itunes on my desktop or laptop as an option to restore with? (short of hooking up the old phone which won't power up?)