Obviously we only have unreliable narrators to go by - by design neither Yandel or Gyldayn are infallible. But going off the history we know of the Dance, Tyrion's hypothesis regarding the death of Byron Swann makes no sense.
Syrax was NEVER IN THE FIELD! She flew Queen Rhaenyra from Dragonstone to Kingslanding, had the unfortunate incident where Prince Joffrey tried to ride her to the Dragonpit to save Tyraxes, shrugging him off to fall to his death - and then was killed herself by the mob at the Dragonpit (not a situation where a knight could try to sneak up on her with a mirror shield).
I suppose we could hypothesize that Byron Swann was a Green knight left in Kingslanding with the recuperating King Aegon II and his family when Rhaenyra came to the city, and attempted to slay the Queen's dragon in the Red Keep while the King and Clubfoot escaped... but such an oversight of specific circumstances seems unlikely for Munkun given his own historic proximity to the events as a member of Aegon III's regency council. As well as being important details that seem unlikely Swann's squire would omit from his letter to Swann's daughter.
If I could offer an alternative hypothesis that seems to fit the timeline and facts we know. Byron Swann was part of the Green forces that marched on Harrenhal under the Kingmaker. Then the following things happened...
1) Kingslanding was taken by Rhaenyra and Daemon, and with King Aegon II MIA presumed dead, Ser Hightower (the Hand) executed, and Queens Alicent & Helaena imprisoned.
2) The Lannister forces were defeated at the Battle of the Lakeshore
3) The Kingmaker was slain and his forces scattered at the Butcher's Ball
Ser Byron can be forgiven then for believing the war was over. He'd likely feel differently if he'd been in the Reach with the army of the late Hand's nephew Lord Ormund Hightower witnessing the heroics of Prince Daeron the Daring, but in the Riverlands the Greens had been crushed, and rather than commanding anyone or acting the hero like his little brother, Prince Regent Aemond was using Vhagar only to burn and terrorize civilian targets throughout the Riverlands.
Ser Byron may have been an unwavering true knight loyal to his king - but for all he knew his king was dead, and a new queen now rightfully sat the Iron Throne. Surviving the Butcher's Ball he'd be without an army in the Riverlands while it was being terrorrized by Aemond and Vhagar. What better way to be true to his vows to protect the smallfolk AND prove himself to his new Queen than to put an end to such a murderous rogue and slay the great wyrm?
So he tries the mirror trick. And he burns. And then the Treason of Tumbleton happens, and the storming the Dragonpit, and Rhaenyra dies on Dragonstone - and the war has turned in a way that the late Byron could not have forseen, and his actions intended to be continuous loyalty to the Iron Throne are left looking like those of a turncloak. So his squire lies! And says it was Syrax and not Vhagar who slew his liege, protecting the late Ser Byron (and by extension his own) good name and honor in the process.
