There is a path that makes the Italian faction more desirable, grow faster. After some runs I figured out that you annex Yugoslavia and Greece like normal, but then justify and conquer Hungary instead of waiting for the France war. It seems that the moment I take Hungary and become Romania's neighbor, I become their preferred choice. Justify and conquer Spain too, because you can. After taking Greece and France, turkey tends to want to join you instead of the axis. The moment you make peace with the allies and end up bordering Venezuela they request to join. If you were working on switching the United States to fascist, they request the day after you annex Canada.the AI most of the time refuses to join your faction and goes for the Axis.
tl;dr I am around 80% confident there is neighbor bonus that can help you, not 100% sure because it could be a war vs peace modifier that switches the choice. If I make sure my Italian faction is more powerful than the axis AND I push my border to Romania, they seem to join Italy like clockwork. It's only when I pursue other targets that Romania seems to jump into the axis.