My MH is the same way sink in bathroom dumps into black tank.
I travel solo and the first time I stayed at a CG that did not have full hook ups I was filing up the black tank 'real' quick.
What I did was stop using the bathroom sink for washing hands, brushing teeth etc. Made a big difference how much longer I could go between having to break camp to dump black tank. Sounds like a small amount of water but over time it 'does' add to filling the black tank. Add a couple of kids to the mix and it starts making a 'big' difference.