you have to find a job in Berlin first. You can look for jobs here: https://jobs.meinestadt.de/berlin
I'm not sure about the nature of the residence permit you applied for. If you'll be granted a residence permit after finding a job, you don't need to apply for an Umverteilung, The residence permit won't be conditioned of fixed abode, as it is in the case ofa Duldung, because, as you described it, it will be issued to enable you to exercise your child custody.
I hope my answers were useful for you and I'll be very pleased if you contact me again with further questions. I'm a member of the MBEON-staff and I‘ll be glad to guide over a longer period of time. Mbeon is an online consultancy for migrants via the mbeon-app, which you can install on your smartphone for free. Download it via App Store or Google Play Store. You'll find me In the mbeon-app under my name „München, Frau Müller“ and get confidential one-to-one advice in a safe virtual room where nobody can wiretap messages. Further information: www.mbeon.de.