After there complete discharge, it certainly needed helped figuring out how long to recharge – it kept telling me another hour and 30 minutes or so for a while. I hope now it knows as it is useful to have an idea how long it’s going to take.
Putting these screws back in was the only problem I ran into – I had to realign the battery a few times before they went in easily.
I would start by doing About this Mac and checking the model identifier in System Report. If they are the same (could be MacBookPro9,2 or MacBookPro9,1), your chances of a straight swap working are higher.
In general with different hardware spec you are safer doing a complete reinstall (you may be in luck if they are the exact same spec other than the screen size). Before you start, make a backup (I prefer Carbon Copy Cloner – it can make a full and bootable backup).
Put the SSD in the 15” and see if it works; if not, reformat and then use Migration Assistant to copy the backup onto the SSD once it’s set up for the newer Mac. This should work if you do a complete backup.