French titre de séjour visa apply for German work permit

[hellofrancais]

Dear all,
I am Chinese but I live in France and hold a valid titre de sejour (a residency permit in France allowing me to live here and search for a job). I got a job offer from Germany and I am starting to prepare my work permit application process. However it has been very difficult for me to get a work permit appointment with the German embassy in Paris so the employer suggested me to look into the ways for me to go into germany, register myself there and apply for work permit from Munich.

So my question is, since I am Chinese with a French titre de sejour residency permit, am I allowed to live in Germany with this titre de sejour? As I need to register myself in the foreigners office in germany, before I can send my application of the work permit locally (I have to prove to them I live in Germany).

thank you all for your help in advance!

[hellofrancais]

To give you more information on me and my situation:

1) the company would require me to move to Munich physically for an indefinite employment contract.
2) I know I could apply for the work permit from the German embassy in Paris, but the nearest appointment I could find is in Oct... which is already too late for me and the employer.
3) I have been living in France since 2019. Got my titre de sejour (a student APS visa) in May 2021 and it is valid until 2022 May.
4) the German embassy or other authorities who are responsible for the work permit, will, in general, ask you to apply for work permit in your country of resident. Due to covid and the restrictions, it is not possible for me to go back to China and apply for the work permit there, so my only choices are applying from either Paris or Germany.