Sorry to hear about your battery change problems. But...
You can buy "000" size philips head screwdrivers as part of a cheap "precision driver" set for less than $5 from hardware stores where I am. Or Amazon, Ebay, etc. Also, I'm almost certain there is a way to replace the battery and put the "capsule" back in without damaging the watch so you couldn't reassemble it.
Jewelers do it all the time. Also, I have over a dozen old Casio watches here with batteries, all with varying difficulties of taking the back off. I don't own this A168 model, but I do have the very similar (plastic cased) F91W-1 that has a similar time module (593 for it vs 1275 for the A168, only difference is the backlighting type) & battery (CR2016) as this one. None of them--including the F91--have been unable to have the battery replaced and reassembled to "original" condition by me.
Unfortunately it seems like your episode was unique. Again sorry to hear, but I wouldn't generalize to say that you can't replace the battery in this model if its done properly.