As I see it the following are the differences, but I am not claiming certainty.
Divisive is something or someone that causes disagreement or antagonism between people (for example in Britain, Brexit) It is about causing division and separation.
Controversial is also someone or something that causes disagreement or debate, but is not about causing division. It is more about arousing discussion of opposing opinions (For example - Trump saying 99% of Covid-19 cases are totally harmless).
Contentious involves or is likely to lead to debate or controversy, whereas controversial intentionally or not arouses debate or controversy.