I agree with OP that there are different nuances. I suggest pro, or more commonly pro-, relates to the mind, to an opinion held. Support is more practical, related to some action taken, or which could be taken in future.
If someone says he is pro-Brexit I understand that he wants the UK to leave the EU.
If someone says he supports Brexit I think I could ask him in what way. Assuming he is a resident of the UK, and a Commonwealth/RoI citizen, then I would at the very least assume that he had made the effort to vote for it. Perhaps he may have delivered leaflets, or attended a demonstration, or even tried to convince friends or colleagues. Whatever he did, I would think as a supporter he did something, or at the very least would be willing to do something if certain circumstances arose.
There is no such implication that someone who is pro-Brexit has done, or ever would do, anything practical about it.
In the case of a supporter of a football team she may attend games, or cheer them when on TV, or tell friends they are her favourite. Exactly what a supporter does is not the point, but they do something, whereas someone who is just pro a particular team simply wants them to win.