As far as I understand, you would like to use a channel-wise convolution with a “per-channel” weighting?
Using groups=5, each input channel will have its own conv kernel.
The dot products of each kernel will then be summed to create a scalar input.
Now, instead of summing the dot products, you would like to multiply each one with a scalar (and add a bias), right?
Well, I’m not sure, if you will get any benefit from this, since it’s a linear operation.
Assuming each kernel performs the following operation:
out(N, C) = bias(C) + \sum[ weight(C, k) x input(N, k) ]

Now you could add your weighting like this (let’s name the scalar a and the bias b):
out(N, C) = bias(C) + \sum[ weight(C_out, k) x input(N, k)  * a + b]
out(N, C) = bias(C) + \sum[ a*weight(C_out, k) x a*input(N, k)  + b]

In my opinion, both the scalar a as well as the bias term b can be learned by the conv bias and the conv weights.
Assuming the model would start with kernel weights of ones and the perfect weighting for the kernel weigths would be [1, 1, 1, 1, 5]. Using your approach, the model could learn these weights, but on the other side, the model can just learn to scale up the kernel weights and compensate the bias.
It’s similar to stacking some linear layers on top of each other without a non-linearity.
f = w1 * w2 * x can be written as f = w3 * x.
Could you explain a bit more about your use case?
Maybe I’m missing something.