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Battlemaster. I consider the champion path to be a red herring.
That equals a chance increase of 5%, and depending on the items you use you can get what - a maximum of 15% critical hit chance? It is not too shabby if you assume using three to six attacks in a round. But a critical hit is just a double damage dice still, while Battlemaster gives you flexibility and you still can have a 10% crit chance easily.
Battlemaster gets additional manouvieres at level 7 and 10.
Champion gets increased jump distance and ability to a half their proficiency bonus to any STR, DEX, or CON check that they are NOT proficient in at level 7, and at level 10 get to choose an additional fighting style.
Don't underestimate those later benefits. 2 fighting styles open up quite a few possibilities. Shield and Defense for the tank, a weapon fighting style and archery to be a dual range threat, etc.
I would still lean toward battlemaster, but don't sleep on champion either. They make excellent fighters as well, and the gap isn't all that big between the two.
It is worth noting that in 5E critical hits only affect damage dice, not flat bonuses. They're great for rogues with multiple damage dice from sneak attacks or for paladins who get an opportunity to land a double-strength smite... but e.g. if you're a fighter getting +8 damage from strength c/o an Elixir of Cloud Giant Strength and +10 from Great Weapon Master, that +18 damage won't be doubled on a crit.
For what it's worth, a barbarian recklessly attacking has advantage (well, barring a source of disadvantage) and crits 9.75% of the time without additional modifiers, almost as much as a champion w/o advantage. Add a single item which improves crit range by 1, and it shifts in favor of barbarian's consistent advantage (19% crit rate vs. 15% for a champion w/o advantage), and the barbarian will eventually get an additional damage die on critical hits, too. Of course, reckless invites attack, but a raging barbarian can probably tank the damage decently, and killing things faster is usually the best damage mitigation anyway (I mean, it's expensive slot-wise throwing around Globe of Invulnerability...).
Besides that it's not a good sub class and battle master is infinitely better
Are the damage modifiers given by battlemaster dice doubled on crit?
The ones that add damage dice (like menacing attack, pushing strike etc) will double on crits
Note that you can get a couple of BM dice with a feat, and have a little of both worlds.