How to use question mark after a name with an exclamation mark at the end

Situations like this are somewhat obscure and there is no clearly correct answer. Not unless you are under strict guidance to follow a specific style guide, and there is no room for interpretation. If that's the case, then you have no choice but to present the sentence as prescribed, regardless of how strange it might look.

As it's not the title of a book or movie, nor is it being used as a reference in non-syntactical representation, common style guides would say that it should remain in normal roman type without quotation marks.

But even if you do put it in italics or between quotation marks, the sentence could look odd anyway:

What did you think of AnimalsUnite!?
What did you think of "AnimalsUnite!"?

You can always choose to rephrase the sentence:

What did you think of the AnimalsUnite! company?

As per one of the comments that had been given under the question (and assuming you are not prevented from doing so for other reasons), you could even choose to drop the exclamation point in the name and just use a question mark instead:

What did you think of AnimalsUnite?

Barring a directive that you must follow, there is no objective guidance in cases like this. Simply do what you think makes the most sense. (And be prepared to defend your choice if you are questioned.)