I think it is a tad unfair to call him "stupid". Sure, he had an inferiority complex that he countered with a degree of egocentrism and dick-waving. But was not an idiot. Just a man whose emotions got ahold of him faster than his wits.
Under a completely different upbringing, say one where he was assuredly going to ascend the British throne and thus never came under the influence of the Prussian aristocracy, he may have been inclined towards an entirely different manner. One wherein he would have been more palatable to the British people.
I assume that, in order to keep the throne separate, the Prusso-German successor would be William's younger, more level-headed brother, Henry. I'm also assuming that Frederick III lives somewhat longer here. I'm not quite clear on the timetable you'd like to see.
I think the most plausible is thus:
1861- Prince Albert and several of their children dies in typhus outbreak in London. Victoria is devastated. Only the Queen, Princess Victoria, and her brother, Prince Albert Edward of Wales, survive the disaster.
1864- Albert Edward's wife dies in childbirth. The Prince of Wales commits suicide in despair. Succession crisis now looms.
1865- Parliament approves William, son of Princess Victoria, as heir presumptive, provided that he be raised and educated in Britain with his grandmother starting at age 10.
1869- William bids goodbye to Germany and his parents to live at the court of the British crown. He is raised, by all accounts, to be a liberal and gentle prince throughout his youth.
1870- Germany unified, William's grandfather declared Emperor. William's father made German Crown Prince.
1879- Emperor Wilhelm I of Germany is assassinated, succeeded by Crown Prince Frederick. Frederick names his second son, Henry, as Crown Prince. Frederick begins a liberalisation in Germany, much to the chagrin of Chancellor Bismarck. William is approved as heir apparent in Britain, and invested as Prince of Wales.
1892- Frederick dies, succeeded by Crown Prince Henry as Emperor Henry VIII, including the Holy Roman Emperors in his numbering to legitimise the connection of the present German Empire with mediaeval Germany; Bismarck resigns.
1901- Victoria dies, succeeded by William as King William V.