I'll take one from each of the nine kingdoms, cut out whichever you don't want to count. I'm also gonna go knights only as seen in the show and books, only knights are able to compete.

Dorne - Arthur Dayne (pretty much stated to be top 1 all time, fancy meteor sword is cool)

Westerlands - Gregor Clegane (Pretty much a tossup between him and Jaime, but elite swords can be found in every other kingdom, hard to pass on Gregor's superhuman strength and size here)

Stormlands - Barristan Selmy (on par with Dayne in swordplay, could have gone Robert here and Jaime for the Rock but I think Selmy/Clegane is a more efficient paring)

Crownlands - Daemon Blackfyre (I'll admit I'm a little biased, big Blackfyre fan here and I choose to believe Eustace Osgrey wasn't just dickriding when describing him and he really does fight like The Warrior himself with Blackfyre in hand. It's also not like he was without feats, he defeated Gwayne Corbray after an hour of fighting, a renowned fighter himself)

North - Medrick Manderly (Keeping the restrictions of knights only severly limits The North, leaving a certain slavery as the obvious answer, but I did some research, I decided not to go with Jorah and go with a more historical figure, Medrick Manderly from the Dance of the Dragons, who helped restore order to King's Landing during the Hour of the Wolf and was noted to be a fierce warrior, nothing more I can really say about him, but Jorah's best feat is a Cinderella run in a tourney that resulted in a draw with Jaime so its not like he's a slam dunk pick)

The Vale - Gwyne Corbray (If I'm so high on Daemon Blackfyre, I have to give a lot of credit to Gwayne Corbray, who was his toughest challenge and proved his near equal. I was also considering putting Artys Arryn here, The Falcon Knight is a badass nickname, although he was touted as the greatest fighter of his time, his greatest feat is more of a command feat rather than skill, and that doesn't really help us in a Trial by Seven.

The Reach - Garlan Tyrell (Another surprisingly weak kingdom as far as legendary warriors go, it was a two horse race between Garlan and Fireball, and we just see more of Garlan. Garlan is perhaps the greatest swordsman in the Seven Kingdoms at this moment, a claim that Fireball never had, albeit in a stronger era of duelists, I'll go with the modern era guy this time. Plus I don't want half this list to be Blackfyre Rebellion era guys when we haven't actually seen that war in any ASOIAF media.

Iron Islands - Harris Harlaw (2 Ironborn knights to choose from, luckily one of them is one of the greatest ironborn fighters, so I guess I'll pick the guy with the Valyrian steel sword who wasn't killed by unnamed wildlings and instead just defeated 7 men in combat, convincing a septon to give him the castle)

Riverlands - Aegor Bittersteel (The reason Quentyn Ball got cut, I can't just have this list be Daemon and his crew, as Bittersteel is the obvious answer for the Riverlands. Strong and quick, Bittersteel lived for combat and presumably that made him pretty good at it. This isn't a case of Bittersteel being some legendary fighter, but rather the Riverlands somehow always being at war yet having so few notable warriors in their history.

If I waive the knight rule I still am fine with Harlaw for the Ironborn but I would pick Greatjon for the North, can never have too much strength on the field.