You did not mention camera/lens setup, but curious on that. You did not mention version of C1, as older versions of C1 do not IMO work as well on the LCC process as 7.x and the wide angle LCC correction.
In low lighting situations where I have a CL-PL on, I will often remove the CL-PL so allow a faster exposure for the LCC, i.e if I am shooting at 1 to 10 seconds already, I don't want to be at 12 seconds for the LCC. In normal light, I will leave the CL-PL on and adjust the shutter speed accordingly.
You can most times not have to shoot a LCC after each shot. I tend to take a LCC series center, right, left shifts then not take anymore unless the lighting changes dramatically or I change out lenses.
The best way to judge an LCC during exposure is to look at the histogram. A well exposed LCC should have a narrow band about dead center. You can be OK if it's just a bit left or right. On shifts, this band tends to be a bit wider. If you are shooting a Phase IQ back you can also check the exposure warning, as if you have over exposed the LCC, you will see a red center.
As I understand the LCC process on a tech camera shot, it should not be working on the exposure of the shot as much as the color casts. On a shift, then yes the LCC will do a lot to even out the exposure of the shift i.e. left or right sides being darker. On certain lenses/cameras you may see a slightly brigher center exposure on a center frame and the LCC can help to balance that out.
I have never seen a properly exposed LCC change the exposure of a shot as you mention.
As for underexposing a LCC, I would rather have a slightly underexposed LCC than over exposed one. Capture One does a very good job of working with the LCC to get an even exposure. Noise in a LCC, should not transfer to the image. C1` is using the LCC to judge the issues of color cast and even exposure (on shifts) against the original image.