I am not sure in how far the 2005 absence was a consious choice and not a coincidence/ oversight. At earlier papal funerals most Northern royal families or even republics did not send the head of state either, nor another member of the royal family. By 2005 things changed, but perhaps the court did not forsee this. This was corrected aftrwards: the investure of Pope Francis was for example attended by the Prince and Princess of Orange and pope Benedikt's investure was attended by the Prince and Prime Minister Balkenende.
From the independance until the Napoleonic times the catholic regions -the so-called Generality lands- were run almost as a colony, not represented in the States General and ruled directly from The Hague. These areas were re-conquered from the Spanish but in the interlaying years the population had been subjected to/ benifited from the counter-reformation and for the most part became catholic again. The -admitedly flawed- logic was that as catholics they could not be trusted to be loyal to the state ánd unlike the rest of the country they had allowed themselves to be occupied by the Spanish while the upright protestants did not.
Although there were no religious wars in the 19th century and onwards the relations between the protestants and catholics were not always cordial and marked by a deep protestant distrust coupled with various popes in Rome trying to strengthen their grip on catholic life and Dutch catholics undergoing a process of emancipation. In the North catholics were only allowed to build open (visible) catholic churches after the change of the constitution of 1848 for example. Before that they would have to be more discreet. For a long time catholics (and Jews, lutherans, baptists etc) would be kept out of the upper elechons of society/politics. The first catholic Prime Minister would only be instated in 1918.
King Willem I toyed with the idea to merge the protestant church of the North and the catholic church of the south into one state church. This met with great resistance. The break-up of the United Netherlands was obviously celebrated in Belgium. But apart from the court, it was also a relief to the North, where the protestants were happy to be the majority again. The 'ungrateful Belgians' were blamed for it all and -not entirely without logic- many saw a papist plot at the root of it.
Willem II tried to come to a concordat with the Vatican, but he was hindered from achieving that by protestant groups.
Willem III became a figurehead for the protestants. In the 1850s he forced a government crisis and the resignation of the government when he supported the April-movement, where protestants signed fierce petitions against the reinstatement of the hierarchy of bisshoprics in the country, According to the writer of the petition the reinstatement was meant to 'make the King [and country] a slave of the pope [...] even more so than Philip II, murderer of William the Silent, ever did. What didn't help was a papal bull of 1853 by Pius IX where calvinism was named a heresy. The King refused to follow the advise of the government to decline to receive the anti-Roman petition, instead he praised the writers. Even after 1853 the bisshops of Haarlem and Utrecht were not allowed to take up residence in these cities for some years. The King's motivation may have been more worldy than religious as he always opposed the constitutional change and wanted to test and strech his powers.
Processions were forbidden by the constitution of 1848, which was only changed in 1983.
In 1885 King Willem III still refused to set foot in the national museum in Amsterdam. The architect was a catholic and the building reminded many of a catholic church and was fiercely critisized. His sister Sophie, Grand Duchess of Weimar funded the restauration of a catholic church on one of her Polish estates. When the renovation was finished she visited it. But she did not exit her carriage and sent a lady-in-waiting to inspect the church as 'a member of the house of Orange does not enter a catholic church'.
Queen Wilhelmina was always considered a rock of protestantism and had a certain reserve towards her catholic subjects. The preconception was always that the catholics were not loyal Dutchmen, as they were loyal to the pope. She resisted the installation of a catholic mayor of the city of Delft out of fear it would leave the royal crypt unprotected. She felt a lifelong connection to the Walloon church of the Hague of the Huguenots and to protestantism in France, as a descendant of Gaspard de Coligny via his daughter Louise, last wife of William the Silent.
This belief was apparently only changed later on in WWII. Initially she received reports about the resistance movements from the protestant vicar Visser 't Hooft. She received no reports from the catholic South, which led her to believe that they were indeed not loyal. This was only corrected in 1944 during the visit to London of a resistance leader of the South. The Queen -drastic as ever- then completely revised her opinion, also influenced by the examplary role of cardinal (still archbishop during the war) De Jong, who was leading catholic resistance. After her return she had many discussions with him and already in december 1945 she awarded him the Grand Cross of the Dutch lion. Upon her entry in Breda it was the archbisshop who was seated next to her in the car, while a protestant vicar took the back seat - much to his dismay. The friendship became so visible that the cardinal refused some royal invitations as he feared rumours about a secret catholic baptism of Pss Marijke (Christina) would start floating around and only accepted them after the protestant baptism. The cardinal later was postumously awarded the Yad Vashem in 2022.
Until the late 1960s or even beyond, much of Dutch society functioned in the so-called 'pillars'. In general people would stay in the pillar they were born: the protestant one, the catholic one, the socialist one, the liberal one. Each pillar would have their own associations, their own political parties and cross-pillar marriages were rare, especially between the religions. Traditionally the protestant pillar was the most Orange-minded, promoting the idea of 'God, The Netherlands and Orange'.
The outrage at Irene's conversion was for a large part focussed on her re-baptism which many protestants found offensive. I believe this is not needed any longer, after the second Vatican Counsil. The outrage in protestant Netherlands focussed on the religious part, and it is sometimes claimed that the conversion was at the base of Irene not seeking parlemental approval for her wedding. This is not however not true. The reason why the marriage would not have been approved by parliament had to do with Carlos-Hugo's Spanish pretentions, not Irene's conversion. The pretentions -not the religion- were also the reason why Queen Juliana could not attend her daughter's wedding.
In 1985 Queen Beatrix was the first Oranje who visited the pope. 'Finalmente' she told him.
In 1998 the (by now tiny) orthodox protestant party still voted against the wedding of Prince Maurits and the catholic Marilène van den Broek. In 2002 they abstained IIRC. The wedding of Maurits and Marilene was oecumenic and Princess Juliana took the communion. Something that was criticised by the more orthodox protestants and catholics alike, as was the case when Bill Clinton too the hosti in South-Africa in 1998.
