It would be the landlord needing to call an electrician in, nothing to do with the electric company (at this stage). Here if a landlord is failing to act we can call in a professional and withold the service fee from the rent.
The inner panel with the oven and other 3 circuits has a total of 100amps, compared to the outside panel only handling 50amps. So you could run enough stuff that wouldn't trip the inner panel, but the outside is overloaded.
But it doesn't preclude a short or some other situation either, like a faulty breaker inside, that should be tripping but isn't.
For safety you need to have an electrician come in.
When does it trip? What things are running when it happens?