What is the appropriate course of action for disagreeing with an authority figure?

I think it depends on the nature of the opinion and the nature of the disagreement. Sometimes figures in authority positions have to make a judgement call about an issue where the formal rules of the situation don't clearly address the situation at hand. I think that in such a case the person in authority should have their decision respected, provided that the decision is based on the relevant principles to the case at hand and is a reasonable interpretation of the facts and rules.

If their decision is just plain not reasonably based on the facts or the relevant principles then it is clearly wrong and the best thing to do is to decide whether or not it is a big enough deal to want to appeal, and if it is, then one should take the matter up with the next highest competent authority, be it their boss or the HR department.

In the case of a monk the correct response would be to bring it to bring it up with whoever is higher in the hierarchy with respect to the disputed matter. For example, if you have a dispute with the Monk in charge of the lost and found or something in a monastery, the correct response would be to go to the person in charge of the building, or to the Abbot if the matter was important enough.

If it is a very serious matter then one should probably go to the Abbot directly and if that fails, then try to find a senior Monk to get the matter brought to the attention of the local Sangha council if their actions are breaking the Vinaya. And of course it goes without saying that if they are committing serious crimes or putting people in danger, one should contact the authorities as well.

But as I said, authority figures have to make judgement calls when the rules of a situation are unclear and as long as their decision is a reasonable one it is better to simply accept their decision. It doesn't make much sense to fight with an authority figure if it's just a case of "They decided A but I think that B is better."

The term "hindrance" (used in the OP) is more usually applied to some aspect of yourself (e.g. doubt, restlessness, ill-will, etc.) than to an external thing or fact (e.g. what someone else says or does).

Perhaps a good answer would be to, somehow, accept or discard the disagreement, or reformulate your perspective so that you view (frame) it differently. Generally, depending on the specific type of hindrance (e.g. "ill-will") there may be some corresponding antidote, for example metta.

The OP says "the vinaya" so, assuming you're asking about monastic rules, this page references vinaya texts, which describe rules which monks accept.

Depending on the topic/subject of your disagreement, one of the relevant rules might be "13. Criticizing or complaining (about a Community official) is to be confessed" on pages 278 through 280 of this document (rules for the official being in Chapter 18, starting on page 272 of this document).

I won't quote from it: two pages is slightly long to quote here entirely, but I don't want to quote extracts from it in case I am wrongly biased in which extracts I select: so, better to read it all, it's only two pages.

On the (slightly separate) subject of "sowing discord" there's something about "dispute resolution" on page 453, which starts with,

In addition, dispute-issues must be settled “in accordance with the majority”; accusation-issues, either by a verdict of mindfulness, a verdict of past insanity, or an act of further punishment; and offense-issues, by acting in accordance with what is admitted or by covering them over as with grass.

It includes the usual kind of "if it's the right topic at the right time" and "if it will be received correctly by the audience" advice:

If the answer to the first four questions is Yes, and to the fifth question No (i.e., the discussion is not likely to lead to strife), he may then go ahead and start the discussion.

It lists "unskillful states" which "can turn disputes into issues."

Reading further, it has advice like,

The Commentary here adds that the residents should first stall for two or three days—saying that they have to wash their robes or fire their bowls first—as a way of subduing the pride of the incoming bhikkhus.

I won't try to summarize it further: it's there for you to read if you think it's relevant.