Heya - I'm normally playing as that Kallari on the other team - sorry 😛 But Kallari has pros and cons too so here's some thoughts from the other side of the coin.
Given that Kallari is an assassin with an excellent suite of ganking tools, I think it's pretty reasonable to expect that she will be able to score more successful ganks on average during the early/mid game. Assuming you are playing a tanky jungle like Rampage, a lot of your power comes from your tankiness, which really shines when the team-fighting starts.
As a Kallari, I really don't like tanks. Particularly tanks with CC. I can unleash a combo but tanks will normally kill me before I can kill them in a prolonged fight since Kallari is fairly squishy (Rampage is the worst). I'm far more interested in the squishy targets who I can kill. But that's where tanks/bruisers can be a problem. If I start to engage on their carry in a teamfight for example, but I get a bolder to my face, Grux stuns me, or anything else, I'll typically die very quickly.
So, in terms of playstyles, I recommend playing as a bodyguard from mid-game onwards. Stay with your squishes and be as quick to peel as possible to make my life a literal nightmare. Doesn't really matter too much if the Kallari is ahead if they are locked down with abilities (also why I don't like playing against Fey).
Tank items such as Stonewall give immediate CC if the you go below 40% HP - that's a horrible to play in to as an assassin!
During the early game, gank and counter-gank as much as is reasonable, but don't fret too much about the Kallari. Try to focus on maintaining really strong farm so that when the mid-game comes around, you are an unstoppable wall which will protect your team. If the Kallari has all the kills, it's not such a bad thing if you shut them down first in the fight.
Also, wards! The natural enemy of Kallari's invisibility. You could consider keeping your regulars vision wards to help your team track the Kallari.