FWIW I made several replacements of batteries and screens for smartphones (HTC, Xiaomi...), not yet for Sony, though.

My comments:

I usually buy the components in Aliexpress but this is just a matter of habits. Note that receiving items from Aliexpress can take quite a long time.

Do not buy the cheapest component (or the most expensive one...) and avoid the batteries which indicates higher capacity than the standard one. This is just marketing and maybe a sign of not so serious seller.

I bought several tools like these for opening the casing:
https://www.aliexpress.com/item/33052488276.html
https://www.aliexpress.com/item/4000067505530.html
But finally I do not use them, I use plastic collected from packaging instead. Being soft, this has the advantage of not damaging the casing. You can use playing cards as well.

Not having the specific adhesive for reinstalling the back plate is not a problem, I use double sided sticky adhesive, about 2mm wide. For example this:
https://www.aliexpress.com/item/32851454528.html
In some places (below the battery, etc), you can put several layers, one next to the other.

Do not overheat the back plate with the hair dryer, you should always be able to touch the parts with bare hands.

Allow plenty of time for doing this, don't do this in a hurry. The first step (removing the back plate) can take 10 minutes or more...

Watch several tutorials and don't start until you're comfortable with the sequence of operations.

Good luck