I don't think either the Kobo or Kindle will handle PDFs (natively) as well as the Sonys. Since I use old readers also, I can't really tell you exactly what you would lose (or gain) by going with a 2016+ Kobo or Kindle. In my opinion, for pure build, nothing beats the Sonys. I do have a Tolino Page with the E-Carta screen and it is "whiter" and sharper than the Pearl. That said, it's not that huge of a difference (my E-Carta screen is only 167dpi, however). I do know that Sony screens are more responsive than Kobo screens (at least that the case on the old models I own). That's the biggest drawback (in my opinion) with the Kobo -- the "double tapping" issue. (Maybe it's just an issue for me?)
At any rate, NewPower99 has the battery kits for both the 650 and T3 for $24 each. You might be able to find them cheaper, but that's not too bad.
https://www.newpower99.com/Sony_PRS_...y%20prs-t3.htm
Or (for $11.29) ...
https://www.ebay.com/itm/Battery-for...0AAOSw4GFbLe7H
https://www.newpower99.com/Sony_PRS_...%20prs-650.htm
Or (for $8.05) ...
https://www.ebay.com/itm/eBook-Batte...wAAOSwGGlaoZaj
Good luck whatever you decide to do.
(You'll also lose the Oxford dictionary on the Kobo -- but not the Kindle.)