Daniel Jaquet in "The plays with the axe in armour of the Anonimo Bolognese (1510-1515)" (2017) uses the examples of illustrations from the 15th century from the Fior Di Battaglia 1410 and De Arte Gladiatoria Dimicandi, 1482-1487 which includes duellers in armour with axes, that title is pretty bang on the nose isn't it? Similarly, Victor Kiernan in "The Duel in European History: Honour and the Reign of the Aristocracy" (1988, 2016 edition) said that trial-by-combat could involve hired intermediaries who would wear heavy armour and distribute wins between each other by bribing each other, essentially rigging matches. So at the basic level, yes, this is certainly possible.

In my own understanding, the standard form of a personal duel in the 16th century was un-armoured or lightly armoured and wearing an arming doublet over your clothes for wearable padding. For example in Kiernan's book there is an illustration from the Bodleian Library of a 16th century Germany training-school for fencers and they appear to be wearing only arming doublets. I think if one fought for one's self you would be more likely not to wear armour oddly enough, because you would want to prove your honour, whereas the armoured swords-for-hire were clearly about winning the trial-by-combat more than anything else. You wouldn't usually have an intermediary for a duel so that leaves the armour as less likely than in the trial. Kiernan describes men stripping to their shirts, presumably meaning the linen undershirts, for duels in order to prove they were not wearing concealed armour; which was considered cheating.

As a general rule duelling was not so common in England in the Medieval times as it was in the 16th and mid 17th centuries when the custom was largely an Italian important and became very fashionable for a time, as Markku Peltonen says in "Francis Bacon, the Earl of Northampton, and the Jacobean Anti-Duelling Campaign" (2001). He includes the amusing detail that men might even duel over being called "thou" in the 16th century because "you" was favoured as the polite pronoun. On the other hand, the trial-by-combat became less common over the Late Medieval period as noted by Keirnan, because of alternatives through judicial investigations or trial by jury. Churchmen were not fond of the violence of trial by combat so they tended to discourage it, sometimes argue for it's banning altogether. Eventually in Western Europe it was abolished or slid into abeyance. After the abolition of trial-by-ordeal for verdicts in favour of juries in England, trial-by-combat became a primarily civil method, and the later fad for duelling would actually have contributed to its decline as that provided alternate and more fashionable way of staging the same contest.

It's worth considering then, that the scenario isn't necessarily likely even though it is possible. Trial-by-combat was more commonly associated with armour, as was jousting. In contrast, there seems to have been a predisposition towards taking the risk of no armour for duels: as being brave and dangerous was the point. Furthermore, personal duels for honour were not common yet in the 15th century, which was late enough that trial-by-combat would have been going out of style. So it's possible, and perhaps other users like u/sunagainstgold might help, but to me it seems to be not the most well-founded idea. The King movie is a bit loose with history as it's an adaption of a fictionalisation (as a creative take on Shakespeare's Henry V). Like a lot of historical fictions, it tries to capture some version of the "spirit" of history but does not real tie itself to specific facts or any developed academic theory.