TL; DR: as a Grandmaster of weaponsHalf barbarian orc, is it worth taking 3-4 (or more) fighter levels to do more damage?
I'm going to play a half-orc barbarian in a D&D 5E pure PHB campaign that will start soon, starting from level 1. I have decided that I will use the Grandmaster of weapons feat, building a great ax (d12).
An average orc barbarian can be a critical hit power, thanks to Wild attacks With Brutal criticism. However, apart from the advantage in attack rolls granted by Reckless attack, there is no way to increase the probability above ~ 10%. This seems a bit unsatisfactory.
I've thought about raising some levels of the Fighter class, maybe after reaching level 5 with the Barbarian, to win Extra attack First. With three levels, choosing the Great gun fighter fighting style and the Champion archetype, I can increase the average damage production and the critical probability (the latter by about + 5%, as far as I understand the mathematics), while getting Surge of action.
This looks great but, of course, slows the road to more Brutal criticism dice and more uses of anger and harm. Also, assuming that I will reach level 20 (17 Barbarian / 3 Fighter) in the campaign, I will not get the last ASI Barbarian (at level 19) and Primal Champion, which means that, at least, I will not gain +2 to the attack and damage rolls awarded by reaching 24 Strength and +2 AC granted by the increase in the Constitution, thanks to Unarmored defense. It could take another level of Fighter (16/4) to compensate for the ASI, but that would also mean one less Brutal criticism dice (since the Barbarian wins the last at level 17), which is against the purpose of such a critical construction oriented to the coup.
I can't decide whether to follow this path or just go for the Barbarian. Maybe I didn't even correctly calculate the increase in average damage production and critical probability, since I really don't know the math.
Will being pure barbarian or one of the multi-class options will result in greater damage?
If you have a different combination of classes to achieve the best damage production, indicate it.