Hello and welcome on board. Don't worry about PoD's plausibility, you'll always find someone ready to crush it.
Speaking of which...
(Could you allow me to answer you in a different order than your questions'?)
Would occitania be more independent?
Medieval Occitania didn't existed : occitan feudal states were like puzzles pieces that didn't belonged to the same box.
Feudal desintegration of the X/XIth centuries was far more important in southern France than elsewhere, and that meant extreme division, weaker great lords, and eventually an almost perpetual war between them.
The XIIth century was a continual war between 4 entities :
Toulouse, Aquitaine (then Plantagenets), Barcelona (then Aragon), and Trencavel.
It's why Capetians intervened in Languedoc before the crusade : during the wars of Henry II against Toulouse, the help of the king was decisive.
So while medieval occitan states were independent de facto, up to a point, they were still under french suzerainity and while french kings actively tried to enforce this link, many occitan lords used it back for their own interests.
It's one of the reasons why the Crusade was as well a religious war, a feudal war from northern lords as a "civil war" : many occitan nobles joined up with Crusaders and at first even Raimond of Toulouse.
What theological implications would come of Cathar preachers freely speaking throughout Europe?
Few.
First, Catharism was an elite heresy (as almost all gnostic heresies).
It was quite limited demographically : a relativly strong urban minority (around 10%) and much weaker in countryside.
In cities, it was essentially touching what we could call a "middle class", aka small merchants, urban nobility and knighthood, but neither popular classes or elites (while, as elites were often intermarried with their lesser counterpart, a familial solidarity played fully).
During the siege of Carcassone, by exemple, the crusaders demanded the population to surrend the identified Cathars : they were five of them.
And if the city refused, it was less because they were Cathars or having a special bias towards them, but because they didn't wanted to surrender fellow citizens.
Then, Catharism was hardly an occitan heresy, but
concerned all of Europe.. It was already a large matter.
What was different in Languedoc was that the political elite was either unwilling or unable (Raimond V did tried to get rid of Cathars, after all) to deal with.
Because of a cultural bias (one of the crusader's arguments was that Jews were allowed to live normally amongst Christians) but also because of aformentioned feudal desintegration that eventually left few power to greater nobles.
if Peter of Aragon had followed the plan at the battle of Muret, could the Cathars have won the Albigensian crusade?
Well, weirdly enough given the name, the crusade (as in political and military matters, not religious of course) ended to have little to do with Catharism.
What was at stake at Muret, was
1) Aragon's protectorate on Languedoc, with Peter II's army
2) Toulouse urban liberties (The revolution of 1189 effectivly made Toulouse its own city-state republic complete with a
contado), with its urban militia
3) Raimond's power, with his ost.
These diverging interests explain why the Occitan-Aragonese army divisions.
Furthermore, Peter II was quite the model of reconquista's knight.
Admittedly, it made him successful in Spain, but that was the issue there : temporizing when the ennemy was ready to break wasn't worthy of him.
So, a good PoD would be to make the bishop that tried to make a truce right before the battle...not doing this : Peter's plan was originally to wait the crusaders make the first move.
Assuming that Peter stop pulling a Lionheart and act sensible, there's a possibility for Occitan-Aragonese army to crush them, and kill Simon de Monfort.
Then, Aragon clearly made his point : not only Carcassès and Razès were under his suzerainty, but the whole
Languedoc.
The issue is that the land was still a
damned puzzle.
Asserting his authority wouldn't be easy : towns acting as independent republics, lords not always recognizing their direct suzerains authority, and an enduring rivalry between Raimondins, Trencavel and Aragon.
Eventually, such an ensemble would have been really hard to be managed without external issues. And you'll have these.
Remember, Plantagenets and Capetians still have interests in the region, and are not going to abandon them sooner (especially Capetians whom star was clearly rising).
Peter's policy is going to be quite simple :
- Get rid of Cathars. They were the pretext of all the ****, and if they continued to be a thing, his victory would have been hard to exploit.
Inquisition didn't existed yet, but it may actually be created earlier ITTL with the support of Aragon.
- Get rid of troublesome nobles and make concessions. Rebellion, reconquest, re-war is likely to happen : Raimond (and others) pledged alliegance to Peter because Philipp II wouldn't have helped them.
With crusaders being history, and an actually closer king, they are gonna reconsider (remember that in medieval occitan tought, reconsidering a pledge was acceptable if situation changed. Not that was considered a good thing, especially by the loosing side).
Either Peter's manage to defeat him, or Raimond VI turn back to the traditional policy of Toulouse with an alliance with Capetians.
A likely outcome would be something
like this (the dotted line points a possible exchange : Aragon could renounce to his holdings around Mende and Milhau, but could ask for Albigès)
Of course, that such situation would last is another thing.
Regional consequences would be important : for instance, likely no great ecclesiastical estates in Languedoc or a possible stronger consular movement.
I know it's not exactly what you were expecting, but I hope that is nevertheless helpful for your TL.