Component (Add-on) is not booked and billed with its own billing period (configured at Billwerk) but until the end of the subscription plan

This is by design on the Billwerk side.
Billwerk confirmed this as follows:

Bei Signup/Upgrade - Order bedeutet „eigenständige Abrechnung“ immer zuerst eine volle Periode.
Es wird nach dem Gebührenintervall geschaut … oder nach der Phase, fall sie kleiner ist als das Intervall.

So there's no way to do it differently.

The only (uncommon and problematic in other ways) might to to use
api/v1/contracts/{Contractid}/componentSubscriptions
instead and solve this manually like

ComponentId: "xxxxx"
Memo:  "test"
ProductOverride: null
Quantity: 1
StartDate: "2024-04-11T22:00:00.000Z"

But that's no real option here, so we should leave this as-is and should always show the current billing PLUS the upcoming (Next ...) billing from the order array below for the next billing period.

See #2 not fixable currently.