Hi Sandy,
the highest German court recently held that you are allowed to apply for driving licence also with Aufenthaltsgestattung and no other documents. You should be able to prove in some way your age.
You can refer the Auslanderbehörde to the following link which summarizes the ruling of the court:
****://***.lto.de/recht/hintergruende/h/bverwg-urteil-3c1615-fahrerlaubnis-asylbewerber-ohne-pass-aufenthaltsgestattung-eigene-angaben-genuegt/
Pls let me know if you continue to have problems with the Ausländerbehörde.