How to Modify Base Attack Bonus

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Bonus HD specify the exact benefit a character receives from them. There is a reason inspire greatness does not mention feats or normal saving throws like companion bonus HD does.

HD follow the general rules for HD unless otherwise stated. Inspire greatness does not mention those things (aside from additional bonuses), so you go with the general rules for HD, which grant you bonus feats 1/3 levels, BAB, save bonuses, skill points, etc.