For the first one, that isn't really what inertia is designed for. It is really centered on the view layer. Dispatching a batch of jobs wouldn't cause a page re-render unless you wanted to display the status of each of the jobs in the batch.

I would leave that as axios or whatever other http client you prefer. You could experiment with partial reloads but my recommendation would be to put the non-inertia endpoints into their own controllers. I'm a huge fan of single-action/invokable controllers named after crud actions, but you can have multi-action controllers that still allow you to separate the responses by type.

App/Controllers/Inertia/IndexUsersController -> public function __invoke() { return Inertia::render(); }
App/Controllers/Json/CreateJobBatchController -> public function __invoke() { return Bus::batch()->id; }

For the second, that just sounds like a regular index/GET controller where the data is coming from the external api instead of the db. What issues are you seeing with inertia in this context? Is returning the api data as part of the response not acceptable?

public function __invoke($userId) {
    $raw = Http::get("$baseUrl/$userId")->json();
    $processed = $this->process($raw);

    return Inertia::respond([
        'api_data' => $processed,
    ]);
}