Snooky
It's not like the GB franchise relationship with BA. GB pays BA so that GB can operate BA style services on routes that BA does not want to/cannot economically operate itself.
LH has contracted Private Air to run a specialist service with a specialist smallish long range aircraft (BBJ). I imagine that it is a fixed term, fixed price contract. If successful, and the passenger numbers increase to that which would fill a full size LH type trans atlantic aircraft, then LH would cancel the contract and do it themselves. If LH pilots are, by then, on strike - one presumes that Private Air will be given a long term contract, and Private Air will be able to name their price.