Assuming I select a secondary natural attack such as a tentacle for the focus of the feat, My question is how would secondary natural attacks, which normally deal only 1/2 Strength bonus to damage, interact with the feat Dragon's ferocity which makes all attacks after the first deal 1 1/2 Strength bonus.
The feat Dragon's Ferocity says.
Benefit: While using Dragon Style, increase your Strength bonus on unarmed strike damage rolls by an additional one-half your Strength bonus, to a total of double your Strength bonus on the first attack and 1-1/2 times your Strength bonus on the other attacks.
The 1-1/2 is an example of using manufactured weapons and not secondary natural weapons. It is not what it sets the damage to, secondary natural attacks would add an additional 1/2 Strength bonus making them deal the full bonus instead of just 1/2.