If the President knew that his allies would let him down so suddenly, he would have handled them with the greatest care.

The sentence in the second clause uses "would have handled", which is a would have + past participle structure. The sentence is a conditional, and since the change is only in the first clause, the correct structure should follow the conditional type 3 structure [if + past perfect, ... would have + past participle].

The correct statement should be " If the President had known that his allies would let him down so suddenly, he would have handled them with the greatest care" or "Had the President had known that his allies would let him down so suddenly, he would have handled them with the greatest care ", making option B the correct choice.