Templar Inquisitor seems rather ouf of context. But what I rant about, is the fact that dwarves in DAI are always born like thugs. There's no way in game to possibly redeem. It's all just headcannon.. Hence....the Dwarf, unless rogue, is OOC.
Now... to just summarize..
Human: You're noble. Lord Trevelyan. How can you possibily be other from Knight in shining armour? Possibly mage... For me... It's 1H.
Qunari: Basically imho the most versatile of all races. Since you're Vashot, you can be anything you want.
Dwarf: Look above.
Elf: Elves tend to be mages. Or Archers. This is due mainly to racial bonus. Some say they are the most lore-centric of all races. Others say it's just out of place and that the lack of elven knowledge is just frustrating.
So... Who except for Qunari is suitable for 2H fighting lore-wise?
Human Noble Archers are very common in Lore, Sebastian? Nathaniel? So I find it makes perfect sense to be a human Archer.
Qunari-are very versatile, i agree.
Dwarves- Well, Carta don't just employ rogues though, Out of the 9 characters that are Carta,, 4 are warriors, Beraht (carta boss), Oskias (smuggler) Rhatigan (Carta Boss and 2 hander berserker) and Roggar (carta Thug) so it makes sense that either type of warrior is accepted, but warriors are in every rank of the Carta are warriors but bosses are more often warriors than Rogues, so a good backstory is being trained to be a boss.
Elves- Ariane, Feneral, Ilen, Paivel, Tamlen are all sword and shield Dalish, some are hunters and also use a bow, but their main weapon is sword and shield and are protectors of the clan. So makes perfect sense to be a Sword And Shield. Really surprised since Ariane and tamlen have been temporary companions.
So Dwarves make sense Lore Wise as being 2 handed, especially since Cadash is a major Carta Family. It could be you were earning your right or whatever, but it does make sense with lore that while a vast majority of Carta are Rogues, most carta bosses are warriors.. The only Rogue boss took over by being the right hand of a warrior boss, so it can make a pretty awesome lore appropriate story.