For the United Irishmen to win they would need to find all the British spies in their ranks and would need a successful French landing with more weapons and troops arriving after that.
The French under Napoleon were not on good term with the Catholic Church.
This could be useful.
I suspect that Napoleon might try to make Ireland a secular state like France.
Non Anglicans in Ulster were also rebelled.
Only Anglicans were consider to be protestants in those day and were a minority even in Ulster.
An end to the tithe paid to the Church of Ireland should be popular the Presbyterians as well as the Catholics.
The Irish would need to allow the Royal Navy to use some ports in Ireland for the British to agree to leave.
I would say the same port as the OTL treaty ports.
After the fall of Napoleon I could see Ireland becoming closer to the America and might even adopt some thing like the US constitution.
A quite long time ago, I wondered about a possible landing on 1796 that went actually quite close to happening and with more french troops.
It may be a better occasion to have a successful (even if temporary) Irish rebellion.