Is there any way of ending up with a really liberal but united Germany?
If the constitutional crisis ends with a liberal victory and Wilhelm's abdication, we are prone to see a much more balanced Prussia. Still with an influential army and a strong monarch, but also a self-conscious Landtag. Considering that these are the 1860s, this should be enough to label it as liberal.
Germany unity would come in a slower, gradual process over the next decades, rather comparable to the development of the EU. The "Zollverein" created a trade-union (with Austria and very few smaller states not being part of it) and created guidelines to converge the national currencies. From OTL-1868 on, the Zollverein had an own Parliament, elected on base of universal male suffrage. It is not hard to see how this could rather smoothly lead to German unification without a war against France. However, this political integration of Southern Germany into the Prussian dominated sphere has to be achieved without the 1866 Bruderkrieg, which would probably not occur without Bismarck. That, actually, is the tricky part.
Or could we perhaps see a Germany remaining divided but with many of the monarchies turning into democratic constitutional monarchies?
Actually, almost all of the members of the German Federation were constitutional monarchies at that point of time. That the franchise was limited in one way or the other was not a German specialty.
As the general trend in Constitutional Monarchiesies goes, it is fair to assume that sooner or later the monarchs will be reduced to figureheads. This could be the case earlier in Bavaria or Württemberg, later elsewhere, last in Mecklenburg
. A longer reigning King Friedrich in Prussia could bring about steps into this direction (without being utopic) which might prove decisive.
Austria(-Hungary) is a different case because it is very hard to make it work on a Parliamentary base. It actually needs very good monarchs to survive.