Not certain what specific years the game covers, but yes in May 1264 he captured King Henry III of England, plus Henry's son and heir Edward (future Edward I) and Henry's younger brother Richard, Earl of Cornwall, at the Battle of Lewes.

To give you some brief, wider context. Simon had originally been a friend of Henry's. He married Henry's sister Eleanor (scandalous at the time as she had become a Nun after her first husband died) and at one point had been sent to Gascony to act as the King's representative. The problem with Simon was that he was arrogant and greedy, and the problem with Henry was that he was weak and amenable to influence from a variety of sources. These sources include his maternal half-siblings, the Poitevins, and his wife and her birth family, the Savoyards. These two competing groups vied against English barons for favours, land grants, money and marriages, and the English barons generally came third in Henry's priorities.

Simon was better off since he was the King's brother-in-law, but he did stupid things like borrow money using the King's name as security without asking Henry's permission. His time in Gascony was brutal but nothing exceptional for the time. But the outcry it caused led to Henry recalling Simon and effectively putting him on trial for what he had done. Simon resented this and spent some time in France before he and Henry patched things up, but it was clear that he was fed up with following the orders of a changeable King. Bad harvests at the time also lead to problems around the country, so when Henry went to Parliament to ask for more money it was Simon who led them to refuse the King's request. There were various attempts by Simon and the Barons (good historical band name) to get the King to agree to various terms that would reduce his power, but the King managed to effectively divide and conquer the Barons (not helped that most of them thought of Simon as an arrogant upstart and hated him as a result), so Simon again left the country.

When he came back in 1263 it was clear that more Barons were fed up with the King's refusal to reform his government. Henry and his wife Eleanor of Provence lived a lavish lifestyle, and owed money for things like trying to put their young son Edmund on the Sicilian throne. The result was that they were heavily in debt, constantly asking Parliament for more money, and the Barons were tired of dealing with it. Civil War broke out after both sides asked King Louis of France (who was married to Henry's wifes sister, Marguerite of Provence, but had a reputation for fairness and justice) to arbitrate, and Louis came down on the side of the King.

After the Battle of Lewes Henry was still King, but he was no longer allowed to make any decisions himself. Everything had to be considered and approved by his "Council", naturally led by Simon de Montfort. This was how he became de facto ruler, a lot of the Council were not inclined to argue against Simon himself as he had a reputation for violence and a bad temper, so it was mostly a case of "what he says, goes".

Prince Edward managed to escape in 1265 and from that point on Simon was doomed. Edward was a good military commander and a stronger figure than his father. He led his troops at the Battle of Evesham, where Simon was defeated and killed.

Sources; The Plantagents - Dan Jones Four Queens - Nancy Goldstone (gives a very good account of this period from the side of Henry's wife, Eleanor of Provence, and her sisters including Marguerite, the Queen of France).