Boehm titrations are the most common method for doing this, though you have to be very careful to make sure you get good results from them. The attached papers should help.

These titrations use different bases and an acid to find the number of acidic/basic sites, and can partially differentiate between different acidic functional groups as well. If you only need the acidic strength, however, a back titration using NaOH should be adequate. The barium oxide shouldn't play too big a role here, but be careful in case you notice it dissolving - I just don't know how it speciates on your biochar.