New iPad mini to restore from a 2015 backup?

I just got a new iPad mini 4 and would like to use a 2015 backup to restore it from. When I connect the new iPad Mini to my MacBook Pro (13 inch, Mid 2012) 2.9 GHz Intel Core i7, with 16 GB of RAM, running OS X - 10.10.5, my iTunes (12.7.4.80) does not see my new iPad Mini.

When I click the preferences, and then devices on iTunes on the older laptop, I see the back up that I want to use, so I know it's there.

I understand that a possible solution could be to upgrade the complete OS - but I am afraid that this won't work, or could end up ruining all files. I should also mention that one of the sole reasons I've held on to this MacBook Pro was so that I could restore my iPad Mini from the back up waiting on this iTunes account/computer.

Is there a way to extract the backup for the iPad Mini from this older laptop, and put it on my newer laptop (running OS X - 10.13.4) so that I could restore my new iPad mini that way - bypassing the older laptop?

The goal, get this iPad mini restored from the 2015 backup somehow.