You probably would've needed to get an address there somehow, yes. There's services that do that for you, the non-shady ones being provided by lawyers and notaries.
[ed.: a tiny office in Leipzig is pretty cheap too, but you'd need to ensure mail doesn't pile up there unchecked.]
Unfortunately I'm not joking about this entire thing. Berlin's underfunded, overstressed bureaucracy is to be avoided like the plague.
[ed.2: to be clear, it'll still not be great. Just less bad.]
You need to demonstrate real substance (such as actually managing/working from there e.g in your case Leipzig ) . If you can't prove real substance they will just shift it back to berlin and you could be also held liable for tax evasion (if there is a lower gewerbehebel )