Your's is an pretty edgy case. All people I know which have been employed dependently by a foreign company, have been employed by a local subsidiary (and I can understand that your foreign employer won't open one just for you).
Just a wild guess: If you are above the minimum income for dependent-employees allowed to be privately ensured, I assume all should be good. But I would still do the "status feststellungsverfahren", or maybe just call them. I think they have a hotline.
(In general many answers/threads here may not be really relevant for you as they are focused around self-employment. What you really want is to be a "dependently-employee-in-a-foreign-company". I have no idea how that works, maybe just do make an appointment w/ a work-lawyer).
With that info, I traveled there for the onboarding (and getting a working visa), only to learn that the local tax authorities disagreed. My boss then suggested to just have me invoice him for the time being, until a subsidiary is created in the EU. Still waiting after two years :-\
I even want to stay in the public health insurance (freiwillig gesetzlich versichert). I actually do not want to benefit from being self employed. I just want to make it legal.