Can I drive my car with a jumpstarter/ powerstation connected to the battery?

No you should not do that , I'm surprised it still runs , Like many things in todays world modern cars can't handle adversity . You should probably use a memory saver if you replace the alternator and have the batt. fully charged so that the new alt. does not have to charge a low/dead batt.

The problem with this type of jump box is there is no protection for electronics so there is a danger of drop outs or surges in volts or amps so you maybe damaging electronics that may not show up right away once you get it fixed properly. I don' get why you have to wait for that part as they are available even in aftermarket reman and new.