First off, WAR is in terms of Wins. The positional adjustment is in terms of Runs. Therefore, your math in getting Betts to a WAR of 16.5 before the positional adjustment is not correct.
The way BR does WAR, in short:
oWAR = batting + positional adjustment
dWAR = fielding + positional adjustment
WAR = batting + fielding + positional adjustment
If you just add oWAR and dWAR, you are counting the positional adjustment twice.
Also, the positional adjustment is based off of 150 games. The adjustment has to be prorated according to the number of innings played.