Hi @Chubby
I’m not sure if your situation has been resolved by now but I wish to provide an answer in any case. Yes, after three months without any response or action on the part of the immigration office, you can file a lawsuit of inactivity (Untätigkeitsklage) with the local administrative court (Verwaltungsgericht).
As we are not lawyers on this platform we cannot advise as to whether this is the best course of action. If you wish to file a lawsuit, then you should get legal advice beforehand. A lawsuit, however, will also take time and cost money. Have you tried contacting an organisation that provide free assitance to migrants such as a refugee social worker that provides support in your asylum housing or a migrant advisory service (Migrationsberatung - MBE)?
The only possible issue I can see at present in your case is that you are still living in asylum accommodation. This cannot be held against you, however, as you have requested to move and your partner has confirmed you are caring for your child. Strictly sparking, the permit § 28 is for parents who have shared custody (gemeinsames Sorgerecht) but can also be issued to parents who do not have this.
Best,
Éanna