Additional Fighting Style is a bit weak for Fighter 10
Fighters typically get an average to strong subclass feature at level 10. That is, a feature central to their identity and/or felt many times per day. Very strong but situational features may also appear.
Arcane Archer (XGtE)
We start with an exception. Arcane Archer only gets an additional Arcane Shot Option. Assuming optimal choices, this will be the 4th best shot for the character, out of a total of eight options. Since the uses of Arcane Shot never exceeds two per short rest, I consider this level 10 feature very weak.
The Battle Master's superiority dice increase from d8 to d10 - a small but noticeable buff. They also learn two more maneuvers, which will be the 6th and 7th best maneuvers out of 16. Tasha's Cauldron of Everything introduces seven more maneuvers, increasing the strength of this feature. Added flexibility and (a little) power makes this average.
Who wouldn't like to have part of the Polearm Master and Sentinel feats, but without having to spend ASIs? This is average with most weapons and strong with reach weapons.
Forcing disadvantage on saving throws is really hard to come by. As a fighter, you can do your normal attacks, Action Surge, and cast a spell. With Polearm Master, you can bop them as they approach you to set up for your upcoming turn. Unfortunately, you can only cast 2nd level spells at this point. Also, your spell save DC is probably not great.
At level 10, this is probably weak (or average with Polearm Master). At level 13, you gain 3rd level spells. Being able to attack three enemies, Action Surge, and force disadvantage on their fear saves brings it up to strong.
Psi Warrior (TCoE)
Some really bad things deal psychic damage, though it's pretty uncommon. Being able to end charmed or frightened by expending one of your numerous Psionic Energy dice (no action required) is great. Fighters aren't known for their Wisdom saves and this gives you an easy out. Effective immunity to two common conditions is enough to push this to strong, despite somewhat niche applications.
Rune Knight (TCoE)
Rune Knight learns an additional rune, bringing their total to four. While that technically makes it the 4th best rune (out of six options), most of the runes are quite strong. This also adds another rune invocation per short rest, since each rune is invoked separately. You also gain a small amount of extra damage. Overall, this is strong.
Features that regain uses when you roll initiative are usually weak. Many adventuring days end with the hardest fight, and you don't want to enter such a fight with a small percentage of a resource. That said, Tireless Spirit is actually pretty great.
If you use Fighting Spirit (which now grants 10 temp HP instead of 5) twice before the final fight, Tireless Spirit essentially turns it into a 1/fight ability. I have personally found this feature to be strong all the way from level 10 to 20. In campaigns with 1-2 fights per long rest, this drops to weak.
So how strong is Champion's 10th level feature?
That (obviously) depends on how good a second Fighting Style is as a standalone subclass feature.
The big four (or six)
Archery, Dueling, Great Weapon Fighting, and Two-Weapon Fighting don't synergize well. I doubt it would change the power level of most builds if you granted all of their benefits in a single Fighting Style. As a subclass feature, a second one of these is very weak.
You probably picked one of these up at level 1. For the sake of comparison, Archery is strong as a standalone subclass feature for the right build (though you weakened yourself by not picking it up at level 1), while the rest are weak by themselves.
TCoE adds Thrown Weapon Fighting and Unarmed Fighting to the list of not-very-synergistic styles.
Blind Fighting (TCoE)
As the characters move into Tier 3 and beyond, more enemies will have access to magical darkness and invisibility. You might not use this every day, but it's great when you need it. As a subclass feature, this is weak for melee builds (due to somewhat niche applications) and very weak for ranged.
This has a wide range of possible power levels, since +1 AC gets better the higher your AC already is. For ranged attackers, this is very weak. For melee attackers, it's weak. Numerous AC boosts can push this up to an average subclass feature.
Protection's cooler older brother. Would you like to (effectively) cast cure wounds as a reaction without expending any resources? Of course you would. Interception is a great Fighting Style! It's an average subclass feature for melee builds, but very weak for ranged.
This is mediocre at level 1. It's very weak at level 10. Most high level enemies scale by adding more attacks, rather than having a single massive attack. On top of that, you have to use it before you know if it'll be useful.
Superior Technique (TCoE)
One manuever and one superiority die just isn't enough. At many tables, you will use this feature once or twice per day. Maneuvers are great, but not when used this infrequently. Lack of uses makes this a weak subclass feature.
Prior to TCoE, many Champions had an underpowered 10th level feature. Ranged builds were very weak, while sword & board builds with lots of magic items were average. Some Tasha's options are worth taking even at level 10; however, they still only count as an average subclass feature at best.
Fighters typically get pretty good features at 10th level, so it seems appropriate to give Champion's weak-to-average feature a bit of a boost. If access to TCoE is assured, then granting two Fighting Styles (instead of one) at level 10 may be enough for non-ranged builds. Otherwise, I recommend an additional feature that synergizes with Champion's identity - a hardy basic fighter with extra crits.