One of the players in my party wants to play a multiclass Barbarian/Fighter, and is on the fence regarding which Fighter archetype to choose. He is using a Greataxe (no dual-wielding). He wants to get the most DPR, and he is between the Champion and Brute (from UA). Due to RP reasons, he's not picking Battle Master, as that archetype implies the character studied martial discipline, and the player/character only want to hit things very hard.

With Champion he would get improved critical, which combined with combat advantage and two attacks can almost guarantee a critical hit per round.

With Brute, he would get an extra d4 on each hit, so although he won't be dealing crits all the turns, he'll be dealing a bit more damage.

One important thing: our group is using a house rule so our critical hits deal max damage + a regular roll (for example: a Greataxe deals 1d12+12+Str).
The extra d4 would also be maxed for the crit.

With all that in mind, which option is better for dealing the most damage?

For a better picture:
We are about to hit lv 4, and it's very likely that the campaign ends somewhere around level 10~14.
The character is a half-orc, so he already has a benefit when he scores a critical hit.
My friend and I discussed also which barbarian path should he take (he was between Berserker, that gives him an extra attack each round when he is raging, and Zealot, which gives him an extra 1d6+1/2 brb level as radiant damage for the first attack), and it's likely he'll choose Zealot (my character is a Paladin/Celestial Warlock, so their friendship can provide the RP background for that pick).