FWIW, attached is how the Green TRC of the default Capture One profile for the EOS 1DS Mark III profile looks. Definitely not a simple gamma, and the C parameter of a curve fit is closer to 1/1.942.... So maybe something like that can be found back in the TIFF's metadata tags if saved with Camera profile.
I don't think the TRC curves of the LUT profiles say that much, if I've understood things correctly they are not even used at all in terms of camera profiling. As far as I know if you have an A2B0 LUT the *TRC and *XYZ tags are ignored, as there is both input curves and matrix in the LUT A2B0 tag itself.
And in that case the input curve may only be there to make a mapping trick to get better integer precision (ICCv2 is an old format integer math unfortunately) and the LUT itself may cancel out that curve and apply something new, which you can't see without putting data through the profile. In other words, it's not possible to use an ICC viewer and that way figure out which pre-processing C1 does. I think the TIFFTAG_TRANSFERFUNCTION in the exported profiling TIF is the thing to look for, looking into that now.
Those TRC curves also show an inverted curve, like you say 1/1.942 when a matrix profile actually needs ~1.8 to linearize, so they really doesn't make any sense to me.
The ICCv2 standard show two types of input profiles 1) XYZ+TRC tags (matrix plus shaper curves), 2) A2B0 only. The mix of both 1&2 is some non-standard stuff by Capture One as far as I understand, which may be because they are used both as camera profiles and can also be attached to a TIFF file just like sRGB.icm or AdobeRGB.icm.
If any ICC guru knows how it all fits together, please let me know...