Hey JOHNR
Usually what I look for in these types of repairs (assuming the computer is still running) is unresponsiveness, frequent unexpected shutdowns, confusion on key inputs, etc,. These type of issues are "Logic Board" related. However MacBook's are never wrong when the "Replace Battery" message comes up.
A simple test that will prove it this is a logic board issue is running the Macbook without the battery on the MagSafe charger. If the Mac can successfully run without any issues its a battery NOT YOUR LOGIC BOARD!
Alongside what @brady8812 has suggested on replacing the battery first, is the issue. I wouldn't mess with the Logic Board, as you found out they aren't cheap to replace and install.
Good Luck!