Before addressing the possible effects of extreme weather, I should first address the source which provides this date for a purported battle. The Annales Cambriae (literally, the 'Welsh Annals') are thought to have been composed in the tenth century, since this is when their entries cease. Their sources are, for the most part, a complete mystery. The writers of the Annales Cambriae were not like Geoffrey of Monmouth, who simply invented battles to fill in his 'history'. They created a dry record of what they thought had actually happened. They left most years blank, evidently lacking any significant information to include. Camlann is also mentioned in Englynion y Beddau, a Welsh poem which is thought to date to the ninth or tenth centuries. This does not mean that there definitely was a Battle of Camlann. The Anglo-Saxon Chronicle, first composed in the late ninth century, definitely includes fictional battles. That said, there is no reason to assume that Camlann is fictional or, indeed, reason to assume that two men named Arthur and Medraut did not fight there. It is faintly possible that the authors of the Annales had a Latin written source for the dating of the battle, but the fact that it is called a gueith suggests a Welsh oral, possibly poetic, source instead, as Guy Halsall notes. This would make it unlikely that the date was accurately preserved, as the Anno Domini standard system of dating had not yet become the norm, especially outside the clergy. Early Welsh poetry does not typically feature any form of dating in any case. The Annales themselves do not use it. We also have a reason to particularly doubt the Annales's dating of Arthurian events. The text places the Battle of Badon Hill in AD 516. Gildas, the only contemporary British source for the period, tells us that the battle took place 44 years prior to his writing, and that the generation of political leaders who could remember Badon had since died. Gildas is usually held to have been writing c. 540, as he denounces a ruler named Maglocunnus, usually thought to be Maelgwyn Gwynedd. Some scholars, such as Guy Halsall and Karen George, dispute this approach and suggest that Gildas actually lived earlier. Either way, this would most likely put the Battle of Badon Hill at least a decade before AD 516. There is no reason to expect the Annales to be any more accurate in the case of Camlann than in the case of Badon. It is also worth mentioning that a number of the other dates provided by the Annales are inconsistent with those provided by other sources. As such, we certainly cannot place the Battle of Camlann (which may be fictional) as happening around the time of the extreme weather of 535-536. But this doesn't really answer your question.

Britain almost certainly did feel severe effects from the extreme weather of 535-536. The Annals of Ulster and the Annals of Innisfallen (thought to be both using a single older, probably near-contemporary, source) both report a 'failure of bread' in 536. It should be noted that the original source was probably composed on Iona, so may refer events in Ireland or Scotland. Those same annals do record battles during this time, though no more than usual by their standards. Warfare in Britain and Ireland was both common and regular during this period. In contrast to the Irish annals, the Annales Cambriae (composed long after the sixth century) does not record many battles. But is it likely that Camlann is recorded simply because it relates to the legend of Arthur. We know from Gildas's testimony that warfare was not uncommon throughout this period.

Famine certainly can turn people to violence in the effort to feed themselves and their families. Written records from the later Middle Ages suggest that crime rose in time of famine. This low-level violence, however, is not quite the same as warfare. It is certainly true that food insecurity can be a direct or indirect cause of war. Steppe and Saharan nomads have often fought to gain access to water and grazing, or to seize livestock. Famine can also cause populations to become dissatisfied with their rulers. Examples might include the Yellow Turban rebellion, or the rise of heresy in Europe following the Great Famine of 1315. In the case of nomad warfare, however, the food insecurity is a permanent pressure, and in the case of political unrest, the famine simply contributes to a rebellious outlook in the long term. You seem to be asking if a brief period of famine was a direct cause of war in the short-term.

The accounts of Patrick, Gildas, and Bede make it clear that warfare in Late Antique Britain was, like in later Medieval Britain, carried out at the behest of the warrior elite. We know that British and Irish leaders carried out cattle raids (though the sources here are mostly later, poetic works), but livestock are a very different matter to crops. Herds can be driven, but bundles of barley have to be carried. Large convoys of wheeled vehicles or pack animals were neither achievable nor practical in this period. There was simply no way of carrying a significant quantity of pillaged grain. Livestock could be moved, but livestock did not significantly dwindle in times of dark and wet weather anyway. Moreover, meat was disproportionately consumed by the small warrior elite, who could secure enough grain for themselves from their own peasants.

A famine makes it more difficult to sustain an army in the field. In this period, soldiers could only carry so much food with them, so they typically had to consume at least some food from the local area, whether through purchase, requisition, or pillage. If an army moves into an area, it significantly increases the local population. In times of plenty, enough food can be found to support it, but in times of famine, the local area lacks enough food for so many people. Furthermore, in times of famine it makes little sense to spend time marching and fighting which could be spent planting vegetables, foraging, or hunting. Not only that, but it's more difficult to walk long distances when you're underfed.

It is difficult to measure the effect of a single year's bad weather with archaeological evidence. In the case of the Great Famine of 1315, we have significant settlement evidence, coin issues, and written records to be able to work out if a particular village was abandoned due to a specific famine. For the sixth century, we lack those things. We can work out if a settlement becomes depopulated in the early sixth century, but we cannot narrow it down to a specific year, or even a specific decade. At the same time, pre-modern warfare tends to leave a very limited, and difficult to find, archaeological footprint. In Catastrophe: An Investigation into the Orings of the Modern World, David Keys incorporates Britain into a more general theory concerning the impact of the extreme weather of 535-536. It has been noted, however, that he only draws upon a few sites which may have been abandoned around this time (such as Dinas Emrys), while ignoring other, larger and more significant sites (such as Tintagel and Dinas Powys), which prospered at this time.

The type of conflict that might be more likely during a time of particular food scarcity might be that caused by westward Germanic folk-migrations, with well-armed Anglo-Saxon newcomers seeking land in the less-populated west of Britain. Gildas, however, tells us the opposite, telling us quite confidently that 'external wars have ceased' since the time of Badon, with only wars between Britons continuing. While it is unlikely that there were no conflicts between British and Germanic elites in the 530s, it would appear that this was a time of relative peace. Going, therefore, by the very little evidence we have, it would seem that the time in question was actually one of relative peace.

Primary Sources Patrick, Epistola, Gildas, De Excidio et Conquestu Britanniae, Bede, Historia Ecclesiastica Gentis Anglorum, Englynion y Beddau, Historia Brittonum, Annales Cambriae, The Annals of Ulster, The Annals of Innisfallen,

Secondary Sources Antti Arjava, 'The Mystery Cloud of 536 CE in Mediterranean Sources', in Dumbarton Oaks Papers 59, Ken Dark, 'Catastrophe', in British Archaeology 49, Ken Dark, Civitas to Kingdom, Stephen Evans, The Lords of Battle: Image and Reality of the Comitatus in Dark Age Britain, Guy Halsall, Worlds of Arthur, Nicholas Higham, Rome, Britain and the Anglo-Saxons, Christopher Gidlow, The Reign of Arthur, David Keys, Catastrophe: An Investigation into the Origins of the Modern World,