Thedas doesn't need unity with Tevinter. And thinking it's going to create sympathy with Venatori is an unnecessary leap by about a mile.
When the Avvar start employing bombs, I'll view the situation as similar.
Thedas doesn't need further conflict with Tevinter either. When all hell breaks lose, its wise to make alliances even with regimes you despise. The US didn't have much love for the USSR but allied with them anyways during the second world war. The Tevinter government is not allied with the Venetori. They're not exactly at war with the inquisition either. If the inquisition suddenly send a tribe of barbarians to attack Tevinter, then how would that effect Tevinter's relationship with the inquisition? You can be damn sure they would be pissed. Anyone would be pissed if you send a group of barbarians to attack them. The Venetori are actively fighting against the inquisition. Its really not that much of a stretch to assume that provoking further hostilities with Tevinter could result in some rise in sympathy for the Venetori. Its really not at all surprising. Its happened in real life. The Taliban has gained the sympathies from local people in Pakistan/Afghanistan several times. The reason mostly being unintentional civilian casualties from US drone strikes. The family members or countrymen of the slain civilians don't care if the US wasn't trying to hurt them. They only want revenge and the best way to get that is by siding with the group that is actively fighting against the US.
If you send in this tribe to Tevinter, who will suffer? Not the Archon or his buddies. Its mostly just gonna be random soldiers and maybe even civilians that will die. Even if the Avvar try to avoid civilian casualties, its not always possible. Also, how do we even know that they WILL avoid civilian casualties? Ever seen the show called "Vikings"? As the name implies its about a bunch of Vikings. They have their own set of morals and codes. They have their own sense of honour. Yet, when they go to raid foreign lands, they don't do much to avoid killing civilians. Some of them do try not to harm innocents, but others don't really give a damn and basically kill everyone in sight. Just because bombs are not being used, does not mean that civilian casualties will suddenly be avoided.
And seriously though, what exactly is the justification for attacking Tevinter when you already have vast hordes of enemies trying to kill you? That's like if a huge war were to break out in Western Europe and some genius suddenly decided to attack Russia to make things even more screwed up.