The LCC only works with the compressed .mos files, which are not compatible with lightroom.
I have the same problem. Lightroom is my main catalogue, but I do not use it for development of .mos files.
Capture one does a great job at that, and I like it that the LCC file merges with the original, and can be reset again.
But that leaves me with files that will not show up Lightroom, so the archive is not really complete.
I am trying out Media Pro as my main archive, but to me at the moment it still has to many flaws.
It feels like I am stuck in the middle, and still trying to find the right solution or workflow.
If Phase One were to let the uncompressed .mos file work with the LCC-correction files everything would be ok, and the files would show up everywhere.