SMC errors are common even if the board itself is not at fault. It's much more likely to be that the SMC is not hearing from one of the many sensors throughout the computer, or one of the sensors is getting bad readings. Agree that it's possible your battery is at fault based on the power management error.
Some parts of the power management system in this model are on the battery board, which is separate from the logic board.
If you've already gone through all the trouble of replacing the logic board, I would see what happens if you try to run diagnostics with the battery intentionally disconnected. Or at least the battery data cable. See if you get different errors. If you just get a battery not detected error in diagnostics, seems like the battery is likely to blame. You may still get SMC errors (because there will still be sensors it's not getting expected values from), but see if it resolves the Power management one.
EDIT: Battery data cable in question is circled. The big fat screw with the arrow pointing at it is for the actual power from the battery coming into the board. All the little cable does is report things like presence, charge levels, temp, etc….