Not really. The Bureau of Diplomatic Security and its representative at post, the Regional Security Officer (RSO), make the call based on intelligence information and threat assessments. Although the Ambassador is the President's personal representative and is in charge of all agencies at post (except operational military commands), he is limited in what lattitude he might have in terms of security. In some cases, Ambassadors are directed or overruled when it comes to such things as reducing diplomatic presence in a country due to security conditions. And DS assigns personal protection details to Ambassadors whether they want it or not.
If the host government actually offered more security due to a perceived increased threat and the Ambassador rejected it, then that is a different matter. I would have a hard time believing such a decision or if one was made, it was in consultation with the RSO. More information would be needed to see what form that offer took place, if it did. And according to the story, the increased security appears to be for the Embassy in Tripoli and not the consulate in Benghazi.
We should also remember the mindset was to have embassy security details with no ammo in their guns as well.Not the greatest protection in the world.