Okay so this is what i've got so far, tell me if any of it is bad or anything.
POD: Ludwig I is cheated/embarrassed/something by his half brother, so says 'screw you', doesn't legitimize that line of the family, Ludwig dies as normal, Bavaria inherits Baden.
Butterflies: Austria gets uncomfortable with a stronger Bavaria. Bavaria cozies up to France.
POD 2: King of Bavaria, Maximilian, almost dies in a hunting accident, shapes up as a ruler, gets imperialistic.
Butterflies: Bavaria goes for Saxony, successfully annexes it. However: this leads Austria and their ally Prussia to attack Bavaria. France and Russia intervene, Franco-Russo-Bavarian alliance win the fight. Bavaria at this point includes Saxony, Baden, and their new gains of the Hohenzollern principalities (from Prussia) and Bohemia (from Austria). Denmark solidifies control of what they want. Russia annexes Ducal Prussia, gives Silesia to Poland. France takes their wants in the Rhineland. UK never loses Hanover, the King puts too many resources there resulting in a less powerful overall presence. Austria and Prussia are pissed and the UK is pretty pissed at French power at the time. Round 2: Coming Soon To A Theater Near You.
If this is all good, possible and dandy, I think I will start the write up.
First, you would have to specify "France". From 1815 to 1875, we have a reactionary kingdom, a liberal kingdom, a liberal republic, a populist empire with liberal tendencies, a republic ... Some of these would work as allies of Bavaria, some less so. And each overthrow of the government might lead to a sea change in the alliance system.
Second, a strong king is not the way to go in a 19 century scenario, IMO. Bavaria and Saxony had historically no serious conflicts and I do not see how they would deelop. A king that simply tries to go conquering neighbors because they are there? Either this Bavaria is an absolutist regime, and that will mean constant unrest in Baden and the Palatinate, or the Landtag is influential enough to stop this.
Frankly, this last post lets me suggest you might want to read up on 19th century pariamentarism. For example, the UK and Hanover were strictly separate politically, fiscally and administratively. The King of the UK cannot simply put British money into Hanoverian coffers without Parliament's assent any more as Elizabeth II can today take money from the Australian budget and but it into Canadian or Barbadian government projects.
In my opinion, you might be able to ceate a situation pre-1848 with Russia, Austria and Prussia on the conservative/reactionary side and the UK and France on the liberal side, with Bavaria and most mid-level German states constitutional and tending towards enlightened liberalism in the 19th c sense.
A war might start when the Holy Alliance (ie Rus, Aus, Prus) tries to remove this liberalism from power in all German states, and the latter receive assistance from France. Since Austria is sidetrack by national uprisings in northern Italy and Hungary, Russia by an revolt in Poland and Prussia by unrest in the western provinces, the Smaller Germanies and France manage to defeat the Austrian and Prussian intervention. Bavaria does *not* take Tyrol (they are incorrigibly conservative mountainfolk) but successfully presses its ancient ties to Julich-Berg, thus they gain the Rheinprovinz, while Westphalia is given to the deposed King of Saxony. (Because I see no way at all to keep Saxony and Thuringia out of the hands of a Prussian-Austrian alliance.)
In the end, there are four Germanies:
a) a consolidated Prussia with Thuringia and the former Saxony, without Rhinland and Westphalia. A struggle between reactionary conservatives and liberal conservatives.
b) Austria, mostly unchanged (at least in the old HRE). Reactionary.
c) the Bavarian-led Rheinbund, with Bavaria, the Hesses, Westphalia, Württemberg, Nassau; formal capital Frankfurt. A constant struggle between Napoleonic progressivism and political catholicism
d) the north with a strong Hanover and its satellites Brunswick, Oldenburg, the Lippes, Holstein and the Hanseatic cities. Politically unorganized, strongly influenced by Manchester-style liberalism.