So a few caveats before I blast you with my bad advice: I don't care for 'correct' color and the general rules around linear, and log. I have no interest in the nuances that differentiate rec709 from sRGB. If I'm delivering video, I make it look nice on my monitor and I'm done. If I'm delivering Log or Linear I make sure that the colors are unchanged from the source and call it a day.
I also don't know what the difference between LogC and Log regular is. I've worked with Log Alexa footage, so I'm guessing I was handed LogC; my treating it as regular Log did not end the world.
So, with all that in mind, you should probably work in Linear Float. To do this without too many headaches (everything other than video is a little headachey) go into your preferences and on the "general" tab, on the left bottom there's one that is called something like "image format". Change it to "Linear" and restart the machine. This'll make it so some of the older modules (keyers, the tracker) will display the image correctly, and set your viewer monitor to default to "linear"
Now, when you're working in batch, use a LUT Editor and choose the preset "Log to Scene Linear". For all your video files you can lut them via the Video to Scene Linear lut, though if you're comping them in, you can skip the middle man and just CC and comp them. The only thing that matters is that at any given exposure your comp looks right. (Shift+E and drag on the viewer to change the exposure) With screen replacements you're going to be working inside the exposed range, so while luting them may be technically correct, it is also possibly overkill.
Do all your comps, then stick a LUT Editor with the preset "Scene Linear to Log" right before your output (or in your output if you like to hide your work) and you should be back to where you started color-wise with your lovely looking comps.
You could do the same workflow but go down to 12bit video, but you may loose detail in the extremes of your picture.
I hope that made sense.