This is a tricky one to guess. As we know, at the end of the war Wilhelm has scant loyalty to his actual rule, the monarchy itself, both in Prussia and in the other states, was deeply disconnected from the people's future vision, political reform was imperative, the vaguely anti-monarchial parties were the majority and Wilhelm was very uncertain as a leader in these circumstances. That said, I do believe he had a liberal, if not progressive steak in him, he was political enough to sense the winds and maverick enough to potentially sail them. Although I think he is riddled with all the classist snobbery of the aristocracy, the German aristocracy held little real political power, the bureaucracy and army had real power, only the weighted electoral system gave the landed class in Prussia relevancy, and I think Wilhelm understood his real power wellsprings from his subjects not the elite. That alone suggests he might partner with the SDP, who surprisingly in its leadership was loyal to a King, and allow a shift in power to the Reichstag. At bottom I think Wilhelm actually wanted the best for Germany and Germans, unlike most dictators who merely crave power and their own vain glory aggrandizement, that concern might guide him to sacrifice power for control, influence rather than power, exactly how we assume a modern state should become more democratic even if the trappings are feudal. I tend to think Wilhelm's best allies are the left leaning majority coalition, they are far more likely to retain a Kaiser than even the army or bureaucracy who in OTL let it go at the slightest suggestion, showing me they were far more interested in their own survival. As he ages I think Wilhelm gets more moderate and more amenable to sharing power, delegating more and assuming a more figure head role.
For me that tends to make the evolved system more federal in its workings, more like the USA, with Wilhelm acting quasi-Presidential, the fellow kingdoms having real say as the states once did through the Senate and both that body (the Bundesrat) and Reichstag asserting "advice and consent". I think the Reichstag works more like the US legislature, less like a Parliament, with Wilhelm having more actual power than the UK King/Queen but less than a US President. I think it will be an odd hybrid. The Chancellor might become a more independent go between, an unelected PM, appointed but confirmed by the other bodies, dependent upon the elected body for funds, sensitive to the states prerogatives, obedient to the Kaiser, more City Manager than Mayor and just as undemocratic at bottom. At least that is the way I hope to craft it because it looks more fun than just another Parliamentary system.
And that leaves us how Wilhelm III reigns. My guess is that he is more playboy than king, even more meddlesome but unmotivated to actually rule, preferring ribbon cuttings and blustery press than daily grind of rule. Overall his reign will erode the esteem the public hold and weaken the institution, and perhaps thankfully so. Ten years of him playing rather than ruling will shift the rest of the real power into institutions that might be better at stability and less subject to personalities. That can give the elected body time to grow up and become a serious arbiter, actually governing by debate, consent and compromise behind all the trappings. I think we can get to a more purely figurehead "head of state" role for the Kaiser come Wilhelm IV.
Now that argues that Wilhelm III will be a somewhat scandalous King, to me more like how Edward VIII might have ruled, but the era might tolerate it, social mores might ease and his drinking, smoking, womanizing, and luxury life style might be tolerated as escapist compared to Wilhelmine conservatism. Without far left or far right ideology getting the center stage, his antics should distract from the serious issues. At bottom the economy will be the real issue for a generation or more. And I think he might make odd friends with Edward as a fellow libertine, filling a similar role to Hollywood despite how the people are still poor and things are still drab. The trick will be for the family to emulate the Windsors, moving back to the traditional bulwark, family values, church, charity, etc. Wilhelm might achieve it enough, III will tarnish it, IV will reinvent it, the question mark is what we think Wilhelm IV will be like, long term it will likely be his reign that affirms how the family is regarded to the modern era.
