Sorry, I just unable to completely understand your point. But in CNN each filter have its on own bias and kernels for different channels. And each filter give you one output. Bias is added after adding all the kernels of filter.
You can get help from following tutorial:

  
      
      Towards Data Science – 1 Jun 18
  
  
    

Intuitively Understanding Convolutions for Deep Learning 7

Exploring the strong visual hierarchies that makes them work

  Reading time: 15 min read