Ah dang, you did everything right, then there isn't really much else you can do except put in buttons from another vintage Tama that will fit
Regarding your screw issue, the Japanese vintage Tamas (and probably even later versions like the Akai, Japanese Connection, etc) use JIS screws (JIS means Japanese Industrial Standard). While a Phillips will 'fit', it won't fit as perfectly tight as a JIS screwdriver will, and every time it slips, it will slightly strip the screw head, making it even more difficult to attempt opening up the Tama again. This was by far one of the best investments I've ever made for cleaning my Tamas and opening them up:
https://www.amazon.co.jp/gp/product/B00E55DL4I/ref=ppx_yo_dt_b_asin_title_o01_s00?ie=UTF8&psc=1 (this is the handle)
https://www.amazon.co.jp/gp/product/B0751K53BJ/ref=ppx_yo_dt_b_asin_title_o01_s00?ie=UTF8&psc=1 (these are the precision bits)
The most important bit is that black bit (the number 0) in the 2nd link. It's a perfect fit for the vintage Tamas (and also for screws under the backplate for the Connection series, and also what I used to open up my Tamagotchi Friends to clean it). I had a similar issue like yours with my Mesutchi which I got last Christmas which came new in box, but one of the screws was just dead stuck in there after all these years (Mesutchi is also a vintage from 1997). Try as I might with the Phillips, I couldn't get it to budge a millimetre, and each attempt was stripping the screw even worse. So, I ordered that set (only took like 4 days to get here),
pressed firmly into the screw, and it was a piece of cake to get it out. Literally something which was beyond frustrating became such an easy task when you've got the right tools to work with. If you want some further reading about this subject:
https://www.tapatalk.com/groups/tamazone/stuck-tamagotchi-screws-guide-t22127.html
Oh, and get those batteries out of there asap once you manage to open it. Although they still work, they might be leaking, which will eventually start causing corrosion on your battery contacts. Best of luck