Color Transform Confusion and LUT?!

I work on a M1 Pro mac and shoot with the A7S3, so for things to look right on export I need to set the color space transform as follows — Input Color Space: Sony S-Gamut3.Cine, Input Gamma: Sony S-Log3, Output Color Space: Rec 709, Output Gamma: Rec 709-A. Now if I export this as a LUT and put it in my camera it looks very dark cuz the camera needs Rec 709 as the output gamma and not 709-A. So should I set it to REcord.709 and export the LUT for my camera?
Also I own Phantom LUT which I'm assuming is by default S-Log3 to Rec.709 so whats the best way to use it on a mac so the the timeline view matches the export?

I would just use the camera's own Rec.709 representation for monitoring and/or recording. Is there a specific reason you want to use a LUT from Resolve? It doesn't offer any benefits. If you want to monitor/record with more creative oriented LUTs like Phantom then just load those.

To still answer your question, yes you should use Rec.709 / Gamma 2.4 as the output.

I would just use the camera's own Rec.709 representation for monitoring and/or recording. Is there a specific reason you want to use a LUT from Resolve? It doesn't offer any benefits. If you want to monitor/record with more creative oriented LUTs like Phantom then just load those.

To still answer your question, yes you should use Rec.709 / Gamma 2.4 as the output.

Well the camera's representation is quite bad and because I use the stock conversion in resolve it makes a lot of sense to be able to see something as close to that as possible. Also with the phantom LUT is know I can just load those in but how to I use those LUTs in resolve considering those are made for S-Log3 to rec709 where as the mac output gamma is rec709a?

I will not go into detail about the idiocy that revolves around video delivery for web/computer based content. If you feel inclined to use Rec.709A you can convert 709/2.4 to 709A in a CST after the LUT is applied when you apply it in Resolve after capture.

Regarding using the Resolve 709 conversion for the camera, the tone mapper DaVinci is applied directly on the input space so the appearance when converting SG3.cine/SLog3 to 709 will be different than converting from DWG/Intermediate to 709 for example.

For concistency sake I would consider saving a LUT from the following setup.
CST: SG3.cine/SL3 -> DWG/I - no tone mapping enabled
CST DWG/I -> Rec.709/Gamma 2.4 - DaVinci tone mapping enabled

Then when working in in DWG/I in Resolve instead of the camera space.
The benefit of this is ensuring rendering is always consistent even if you bring in footage from a different camera. Looks you'd save can be saved in DWG/I as well and be applicable in any project using the same working space.

You could do the same but use SG3.cine/SL3 for everything instead but this could be less desirable as it's not that much considered a color working space. But this would really be personal preference.