Late Carolingian Aquitaine
This one get overlooked a lot while it's probably the best opportunity.
To be honest, we should rather talk about opportunit
ies.
The tentative of Peppin II and III to keep or take back the kingship of Aquitaine isn't without merits and possibilities, but these largely depends on Charles II of WFrancia being deprived of skills or life : Peppinid legitimacy largely was issued from local nobility and from alliances with opponents of Charles (notably Lothar) rather than clear support base. Once Charles does manage to secure WF, he does manage to have a freeway to Aquitaine without Lothar being much able or even willing to do something (basically, early medieval Aquitaine is in dire need of allies).
At this point, eventually, Aquitaine is seen as a kingship constituent from WFrancia : not always a given territory, but rather a title to strengthen one legitimacy or one's successor.
Basically, you need to break any strong Western Francia (with a caveat Aquitaine might become Western Francia)
While Frankish nobles first rejected in the latter part of the century the Carolingian claim to the WFrankish throne, it wasn't a unanimous decision : many princes refused (mostly for their own interest) to acknowledge Eudes I as king for a while, who never really managed to affirm its authority beyond Loire and Meuse.
Ramnulf II, for instance, not only supported Charles II against Eudes, but claimed the kingdom of Aquitaine for himself (it's unclear if he declared himself king, or taking it for Charles)
Aquitaine as an independent entity was then somehow resurrected (altough not at all consistent), and it open a series of PoD onwards on which the opposition between Robertians/Capetians and Carolingians could lead to either a new Carolingian Kingdom of Aquitaine or a Ramnulfid kingdom.
Similarily, for exemple, the opposition of Guilhèm II (the last great late Carolingian duke of Aquitaine) against Raoul (Robertian by alliance).
Note, tough, that the ducal power was shifting from house to house, and that the dismembrement of Aquitaine along great houses (Auvergne, Ramnulfids, Raimondins for the greatest) already began to fragilize the region : maybe it would buttefly away the exinction of the House of Auvergne that led to a really damaging succession war between the other houses and a great feudal desintegration in Aquitainr (that was felt up to the Capetian tekeover), but the kingdom would be in a situation midway between WFrancia and Ottonian Italy geopolitically wise.
Anyhow, after the death of Acfred and the War of Succession of Auvergne, there's not really a way to make an extremely divised Aquitaine a kingdom. So let's go for two late medieval exemples.
Barcelonese Aquitaine (
@phil03)
Phil03 proposed a sort of mangled Aquitaine issued from Muret. I'm afraid it's simply not possible given the context.
Languedoc was, as all Aquitaine, deeply divided in small feudal states. In fact, the Great Southern War that took place before the Crusade significantly further weakened Languedoc not only trough general weariness but also with the rise of smaller nobles and urban republics.
Aragon/Barcelone wasn't welcomed as a cultural brother helping against French (as I see it narated far too often), but as a last resort from Raimondins (traditional foes of Barcelone and traditional allies of Capetians) to maintain their states against Crusaders (it didn't really worked, hence the Battle of Muret) and other local dynasties.
I stress this : the land was a
damned puzzle and even people studying it are depressing over the sheer ammount of local dynasties which existed everywhere but had more relevance there (would it be the shitload of Viscounts of Lescure : some working for Crusaders, some working for Raimondins)
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.
Even with a victory at Muret, Peìre would have to deal with a bad geopolitical situation, with everyone trying to get as much as possible out of the situation : and either abandoning the region to its own shenenigans or trying to clear the mess, it would still be Capetians that would have the better hand (especially as "oh well, they would be further suzerains and Peter is becoming a nuisance anyway")
I'd fully expect a modified Treaty of Corbeil, where Aragon could push the border to not only Carcassès and Razès (that they claimed anyway) but maybe Albigès as well, while the situation would a bit better for Raimondins and Trencavel. Certainly not a transpyrenean kingdom.
Plantagenêt/Lancaster Aquitaine
Again, I'm quite surprised that nobody picked this one in spite of the anglocentric view on this board.
Of course, I'm not talking about the so-called Angevine Empire*, but rather the late medieval duchy of Aquitaine during HYW.
Aquitaine was given up in full holding to Plantagenets with the Treaty of Brétigny, and was several times promised to Lancaster nobles during the civil war between Armagnacs and Bourguignons, by each side to gain help against the other.
It happened that Charles V was a magnificent bastard that managed to break the treaty even before signing it (the exchange of places was never reallt effective) in first place, and that Henry V wanted the whole of the prize, not just Aquitaine.
With the right events (death of Charles V or Henry V at the right time), you could see the Duchy of Aquitaine blossoming as a constituent part of the English kingship until realizing that contrary to Wales, it was a bit too great to be that efficiently supervized from across the sea.
I don't think it would be a separated kingdom (rather a constitutent kingdom) or that it would be really that safe from Valois takeover (while probably more good at this), but you would have a late medieval kingdom of Aquitaine with possibilities of survival for some time.
*Angevine Empire, in spite of its name, certainly wasn't an unified demesne. It was rather a common feudal hegemony on really diverse demesnes, themselves often divided in small entities (especially Aquitaine, that was a true political mosaic). Hence why the revolts of Henry II's sons fit remarkably the demesnes they recieved : Aquitaine, Anjou, Normandy, etc. each with their own identity, their own structures, their own interests.
All of that under the still present suzerainty of the French kings that could play on the feudal piano quite easily.