If a response to a question was "我很喜欢看法国电影," would the question be "你很喜欢不很喜欢看法国电影？" or "你喜欢不喜欢看法国电影?"
I understand your train of thought and it is reasonable: you wish to make the distinction between asking 'whether the person likes French movies a lot', and 'whether the person likes French movies (at all)'.

Chinese allows such a distinction by asking: 你很喜歡看法國電影嗎？ and 你喜（歡）不喜歡看法國電影？ respectively.

Rather, I would say it is the requirement of your test that bounds your options, not Chinese. That is because the form V+不+V rarely (dare I say never) permits a verb that is prefixed with an adverb of extent (e.g., 很); the asker is only interested in receiving a yes-or-no answer (think of that as asking, do you V? Or do you 不+V?). Of course, it is at the answerer's discretion that they add an adverb of extent for the sake of emphasis (i.e., 'Yes, I like French movies, and I tell you additionally that I like them a lot.'). In short, you ask a question strictly in 'yes-no', but you answer it with much liberty.

Moreover, the construction V+不+V is a succinct one, which makes 很喜歡不很喜歡 unideal and odd to the native ear. In fact, one is allowed to only repeat the first character of the verb, as in 喜不喜歡, 討不討厭.

The principle of succinctness is further demonstrated in compound verbs. One repeats only the first verb in the compound and not the rest: e.g., with 喜歡吃 (to like eating), we say 喜不喜歡吃, not 喜歡吃不喜歡吃.


The difference between 是不是+V and V+不+V

Curiosity's answer is perfectly valid in this context, but that does not suggest 你喜不喜歡看 is interchangeable with 你是不是喜歡看.

是不是+V is used primarily for confirmation, and in that there is assumption. 你是不是喜歡看法國電影？ contains the asker's assumption that the other person likes French movies. In fact, 你是不是不喜歡看法國電影？ assumes the opposite, e.g., when the other person is showing signs of disgust. V+不+V does not work in the negative sense, nor does it contain any assumption.

We can use a verb in the past tense with 是不是, whereas V+不+V is relatively tenseless. E.g., 他是不是殺了人？ means 'Did he kill someone?' Whereas 他殺不殺了人？ is ungrammatical. 他殺不殺人？ is grammatical and may mean 'Does he kill people?', as if the asker suspects he may be a killer, an instance where 'to kill' is habitual.