All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.
You did not clarify to us the business you wish to establish in order to earn your living so that we could give a ruling about it.
However, in general we can say that if someone finds a job that is dubious, and another job that is expressly Haraam (forbidden), then he is not allowed to do the Haraam job as he can fulfil his needs by doing the dubious job, as what is dubious is less serious than what is expressly Haraam.
As regards using Haraam or dubious money for building mosques, then this is permissible as we have clarified in Fatwa 115211.
On the other hand, it is not permissible to seek a job with Haraam income even with the intention of giving the income to charity for a mosque or else as clarified in Fatwa 83081. But if someone does not find a way to support himself and his dependants except with a job whose income is Haraam and he cannot find another job, then he may continue in that job, but he is obliged to look for a permissible job. In case he finds it, he is obliged to leave the Haraam job as "necessity is to be given its proper estimate". For more benefit, please refer to Fatwa 124718.
Allaah Knows best.