I think the problem here is the severe misunderstandings many people have regarding "Our Fritz". The man was utterly opposed to a parliamentary system. He was an outspoken critic of the electoral vote for the Reichstag and vastly preferred a bicameral system like in Prussia. He had a massive dynastic arrogance and disparaged the other German Monarchs often as Napoleonic Kings. His relationship with the Bavarian King was as close to hatred as you can come. And this was typical for most of them, because in his own words he would make them kneel before him. A tendency that was known well enough among all of them.
Vanity was one of his great characteristics. There are multiple accounts of him being extremely displeased when people did not recognize him on parades or military maneuvers. He often rode from one end to another, so he would be seen by all.
Some examples of quotes about him by contemporaries or instances indicating such behavior:
Frederick William’s brother-in-law, the Grand Duke of Baden, described him as too soft, helpless, uninformed, sluggish, and spineless. Baron Franz von Roggenbach thought the crown prince was incapable of manly resolve and expected that, as emperor, he would waste time and goodwill on mere ceremonial trifles. Gustav Freytag commented on the “weakness of this peculiar personality,” and expected little more of his reign than “quaintness, courtfrippery and decorations.” All of these damning verdicts came from Frederick William’s circle of “friends.”
Bismarck variously called the crown prince delusional, lazy,stupid, haughty, and keen on flattery. The diplomat Philipp von Eulenburg was repulsed by what he saw as the crown prince’s “autocratic desires clothed in sentimental liberality” and his “personal vanity which values hurrahs, bouquets and gestures of prostration.”According to Bismarck’s son Herbert, Frederick William was characterised by “measureless personal pride and the most complete lack of judgement.” His accession to the imperial throne would be like giving a fragile work of art to a “foolish, bloody-minded child.” Friedrich von Holstein described the crown prince as a mixture of “weakness and pusillanimity.”
Even after they had joined the empire in 1871, the crown prince’s opinion of the “German-Napoleonic kings,” as he sneeringly called them, did not improve: in 1873 Frederick William compared the German princes to wasps whose wings had been ripped off—“as long as they can crawl, they will sting.”
"You know what I think about the three German kingdoms [of Bavaria, Württemberg, and Saxony] which we received from Napoleon during a most shameful time to assure Germany’s fragmentation forever,” he wrote in August 1881, and deplored the “daily annoyance these cabinets, replete with their empty titles, cause to the weal of the Reich.”
In December 1862, for instance, he rejoiced at receiving the Order of the Golden Fleece from the queen of Spain and explained that he “had really wanted this old order next to the Garter for a long time.” Eight years later, the crown prince informed Schweinitz that he was “hopping mad that the Pour-le-Mérite had been wasted on Emperor Alexander, who sends me the fourth class of the Order of St George.”
Alfred Count Waldersee,the deputy chief of the general staff, complained in 1884 that during manoeuvres the crown prince kept chasing from one point to another in order to be seen and admired by the soldiers and the public. The following year, Holstein recorded in detail Frederick William’s anger when even lowly coachmen or hapless sentries failed to recognise him immediately.
He arranged himself with Bismarck and while there was a something like a cycle between his like and dislike of him, he still accepted his skill and saw a clear need of him. Considering their best period was the time at the start of the Empire, it is doubtful that it would be different from OTL between them. The deciding changing point in their relationship was the War of 1866, it removed the initial hostility between them that came from Bismarck's solution to the constitutional crisis and his actions thereafter, the fate of Schleswig-Holstein was an unexpected bone of contention. But from this moment on Fritz realized how his own opinion and goals aligned with Bismarck on some points, his skill and how he had become a part of Germany. It is a less well-known fact that the Crown Prince interceded on Bismarck's behalf to ensure a modest settlement against Austria even though his father and his father's military advisors wanted to enact a more punishing one.
What would be different? A Kulturkampf that would not be stopped or far harsher and escalated from the OTL events. Frederick Wilhelm was severely disappointed when the Kulturkampf was called off and did not want Bismarck to relent till they are destroyed. Additionally, he was opposed towards the concession made to the Southern States and pushed for a more centralized constitution. Furthermore, he was likely the biggest dynastic snob of the era. So much that close friends assumed he would never appoint a Müller, Schulze or Friedberg instead of a prince.
If anyone believes he would be inclined towards a less harsh or neutral stance towards the social democrats, let me disabuse you of the notion. His comments regarding them were in no manner below the more vile comments of his son.
The Battenberg marriage may have gone through, even though he himself was also against it. He was more interested in colonies than Bismarck, so there is the possibility for more territorial gains before 1888. His massive Anglophilia is well-known, but Britain was in the period of splendid isolation, therefore nothing would have been achieved in this direction. Maybe better relations with the Gladstone premierships, but I can't see any great change resulting from these instances. A big difference could be the change in relation to Russia from friendly under Wilhelm I. to worsen under Fritz, to then be restored under Wilhelm II.
Ironically, going by his writings in 1885 he had similar ideas to his son regarding a "personal rule". With turning the chancellor into a mere servant and with the support of the armed forces and that of the nation as a whole, the monarch would be recognized as the only original source of political power and would personally conduct the affairs of the Reich.
Another option would be a short intermission between two periods of Chancellorship for Bismarck. When the left liberals had a massive victory, they could have supplanted Bismarck for one election period, even though they would likely be unable to achieve much against all other groups in the parliament and the Kaiser. He never would have agreed to make the army a parliamentary force nor weaken monarchical powers. So, I see this more like a short happy marriage that would have quickly turned sour.
Overall, the 99-days Kaiser has been made into this dream like figure that would have changed the course of Germany, but under closer scrutiny it is obvious how wrong such an assessment is. And this does not even touch the influence from the most unpopular Queen/Crown Princess/Empress in Prussian/German history.
