All I needed was the right tool!
I bought an SSD replacement kit from Transcend (the JetDrive kit) for my 2012 MacBook Pro 13". The kit comes with all necessary tools (screwdrivers, etc.) and I was thinking that this would be a no-brainer. I was wrong. For some reason, one of the screws holding the lower case (the longer upper-right screw in ifixit's repair guide) refused to budge. I stupidly tried to force my way in, resulting in an almost-stripped screw (see photo). I've tried the rubber-band trick to no avail. I was going to use the superglue trick, but I thought that I would need a replacement screw after that. So, I decided to order a replacement screw set from ifixit. While I'm at it, I decided to order a Philips #00 screwdriver from ifixit, too.
The package arrived a bit earlier than expected, and I was able to gather all stuff I thought I would need—including a superglue tube. However, I decided to give a final try to remove the screw without superglue, using the ifixit screwdriver. The risk, of course, would have been an even more damaged screw head. I held my breath and give it a try anyway.
And... what an anti-climax. Using ifixit's screwdriver, I could effortlessly remove the almost-stripped screw! It was just as if the screw was not even damaged in the first place!
So I removed the screw, proceeded to replace my HDD with the SSD, and replaced the almost stripped screw with one from the set I ordered. Voila! It was that simple!
Using the right tool is indeed the key. I thought I was using the correct tool (it came with an SSD replacement kit, for crying out loud!). But apparently I was mistaken. My advice would be to buy tools from sites like ifixit because they are carefully designed for such repairs.