I'm not quite sure what precisely creates the "biting" flavor/effect you're discussing, but I imagine it's a different mixture of acidity or something (since you also notice it in pickles).
Anyhow, I don't know precisely how to obtain your desired effect, but I do know that there are various fermented milk processes in various countries that create similar yet different final products. The greatest variables tend to be the exact composition of the fermenting bacteria culture (which sometimes is just a single type of bacteria, but other times contains various cultures) and the fermentation conditions (where time and temperature tend to be chosen based on what's ideal for the particular bacterial culture in question). Temperature in particular can change the relative proportion of bacteria strains in the final product, and since different bacteria may produce somewhat different waste products as they grow, the flavors will be altered.
The best way to reproduce your desired effect would thus be to find a "live culture" sample of buttermilk with the flavor you like and then ferment it under the same conditions that were used originally. Alternatively, you could experiment with varying fermentation temperature with your existing culture, and/or proportion of starter to fresh milk and see what happens.