It's using the subjunctive mood. Often when you see words like "insist" and "demand", particularly followed by "that", they indicate the use of the subjunctive. "That" is implied/omitted in this sentence. This changes the verb when using the 3rd person. He will insist (that) she have the power of three. Has is changed to have by the use of the subjunctive mood.

No, it's actually correct and here's the way to see it. If you re-write the sentence thus: "He will insist that she have the power of three with her." You can more easily see why the word "have" must be used, rather than "has." The "power of three" is plural, more than one. Plural = have, Single = has.

He insists she has it = he knows that she has it and he is telling us that it is so. He insists she have it = she doesn't have it but he wants her to have it and he is demanding that someone gives it to her.