My own (limited) experience was learning Chen style Tai Chi (and a sword form) from a sifu, who also taught a "kung fu" to small children (a couple of years older than yours perhaps), and other forms (including diverse "weapon" forms) to teenagers.
I found it thoroughly harmless; and beneficial, good physical training. People told me it helped my posture, a lot. My class-mates were wholesome friends.
It might even have helped to keep me out of an actual fight with a drunk one night (because it gave me experience of staying calm without panic when someone threatened me inside my "space").
IMO it's not meant to be "violent" behaviour: it's meant to be "controlled" behaviour (maybe, even, relaxed behaviour).
Our sifu gave us a few "self-defence" tips (though it wasn't touted as a self-defence class), and the primary lessons were: don't fight; avoid a situation (e.g. accepting an unscheduled competition or lesson with a random person on the street or another school) that could ever escalate. I remember his demonstrating his number one self-defence move: run away (i.e. "turn around and run away", don't try to run backwards). A last resort, only if you couldn't run away, stop your attacker and then run away.
No, it's not about that.
It's a sport; it's self-control; and it's cooperative i.e. performed with willing partners or not at all.
I suppose another possibility might be ... dance? But that idea does worry me a bit: because how careful are dance teachers to avoid injuries? And I'd guess that a martial art could be more useful.
I can't advise on choosing a class, a form, or a teacher, but you might find advice on martialarts.SE or on parenting.SE
And anyway surely kids always engage in "potentially violent" behaviour? A martial arts class must always distinguish between "potential" and "violent".
Something I saw someone 'like' on Facebook recently was If She’s Not Having Fun You Have To Stop:
An important lesson for all children: "A boy and a girl run around on the grass at the park. The boy tackles the girl. The girl laughs. She gets up and runs away. She loves to run. He chases, she turns and they grab each other, tumble and land in a pile, giggling. After a few minutes, he tackles her again and she lands a bit hard. She is bigger and physical, but he more than holds his own in roughhousing. She pauses for a second. Then she laughs again; she’s still having fun.
Dad gets his attention, and says, “If she’s not having fun, you have to stop.”
I suppose we were in that class for the health, fun and socializing, fresh air, self-improvement, learning about self (skandhas): and not violence.
I guess she won't be taught how to attack people (neither now at her age, nor later when she's older).
When we learned to swim we were warned it's dangerous to approach a drowning person: in their panic they might try to climb on top of you to get out of the water, drowning you. It's ignorance and out-of-control panic, not knowing how to swim, that makes them and their situation dangerous.
I'm not saying that a Buddhist should advocate it but I sure don't see it as contrary, any more than athletics or calligraphy or swimming lessons are "inherently conflicted". I suppose beware that it's possible there are some 'cults' (unskilled or unqualified or even immoral teachers, or fellow-students), but I guess that's more a reason to be wise about it than a reason to be deterred from searching at all.
Perhaps I should add a disclaimer, than the beginner's forms of Chen style Tai Chi are maybe such that people might not consider it a "martial art" at all.
Still, the "kung fu" forms that he taught instead of Tai Chi to the small children were scarcely more: the one lesson of theirs I saw was only non-contact forms.
I've no idea how they teach karate to five-year-olds. Here is one introduction to different forms.
I imagine (and I hope I'm not wrong in saying) that it's always training rather than fighting: never intention to fight.