Technically speaking Ireland was already independent in 1800, since the Act of Union didn't come into effect until 1801.
If Napoleon managed to occupy a large part of mainland Britain, he could probably have forced a peace settlement involving the independence of Ireland. This is pretty unlikely, though, given the relative states of the two countries' navies at the time (France's navy still hadn't recovered from the big purge of royalist officers during the Revolution). Alternatively you could have a more successful Home Rule movement in the late 19th century. IOTL the second Irish Home Rule Bill passed the Commons in 1893 (the first, in 1886, having been defeated), but was voted down by the House of Lords. If Their Lordships are better-disposed towards the Bill, you'd have Home Rule in 1895 or so (leaving a couple of years to set everything up).
So if we go with a George III period, maybe the independence involved Ireland sharing a monarch with Britain, but otherwise running most of it's own affairs, maybe even the construction of an official royal residence in Dublin helped to keep the Monarchy present in Ireland for longer than it would have been...
Of course if any Monarch married a Catholic, or converted themselves,
The monarchy had an official royal residence in Dublin. It's called Dublin Castle.
They just didn't use it because the monarchy barely left London let alone going to Ireland.
Really? Because I remember reading that Victoria refused to build an official residence in Ireland at one point.
...Then they would no longer be legally eligible to inherit the throne, and still wouldn't be to this day.
It would have been as a replacement for Dublin Castle. The main reason she refused to was because she had essentially no desire to ever go to Ireland in person, and saw no real need to (hell, she only ever visited Wales during some Royal tours as the heiress apparent in the 1830s.)
“I am very sorry to leave Ireland. I have had an extremely pleasant time,” she said at the end of her last official visit to the Emerald Isle in 1900.