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.
I have the kit for replacing a 4th gen iPod battery and the screwdrivers look like the same spec – do the plastic opening tools work as a spudger?