I'm going to wander off on a completely different tack but one that I think will meet your needs more.
Based on the first paragraph of your question it is clear that you would like to visit a Royal Palace - preferably one used by the Queen - which is furnished and and to your eyes and those of the family looks like a palace inside and out.
Perhaps it is best to not get hung up with the names that building complexes have. What you are looking for is a viewable royal residence.
Windsor Castle checks all the boxes. Windsor Castle has been occupied by the Royal Family since 1110 when Henry I built Royal apartments. Windsor Castle is the oldest and largest occupied castle in the world. It is open to visitors throughout the year.
Founded by William the Conqueror in the 11th century, it has since been the home of 39 monarchs. Today the Queen spends most of her private weekends at the Castle.
It is fully furnished, and feels and looks like a palace. It is one. Be aware that Windsor Castle will be closed all day on Monday 17 June and closes early on Thursday 27 June (last admission 15:00, Castle closes 16:00). The castle will be open but the State Apartments will be closed on June 14, 16, 18. Best to plan around those dates. St George's Chapel is closed on the 14th and 15th of June. They close when special royal events are happening.
The State Rooms are incredible. The Semi-State rooms are never open in the summer.
In my opinion it is the best place to tick all the boxes. Oh yes, they do a Changing of the Guard too. Meghan and Harry's wedding last Spring was at Windsor Castle, in the St George's Chapel which you can visit when it is open.
And the new Undercroft cafe should be open by then (watch this space)...