The problem here is that you're not quite understanding what part of the process of RAW development (I think these were shot RAW?) is down to Adobe and what part of the process is down to Canon. Loads of people don't quite understand and think it's down to the camera
Each camera has a profile assigned to it in LR. The way colours are processed is actually down to Adobe (or Phase one etc...) NOT Canon. Canon set the white balance - that's their part of the equation. The WB algorithm has changed between the two cameras, but you say you set the WB using a grey card so that takes that out of the equation. The sensor does make a difference to the profiled colours, but much less than the profile itself.
I calibrate my cameras using a colour checker passport. The LR profile for 5d2 is MASSIVELY out. The LR profile for the 5d3 is MUCH closer to real. So, it's the 5d2 profile which is wrong and the 5d3 profile which is right.
However, you like the look of the 5d2 profile.
If you want to get the same result as the previous camera, I'd recommend getting a colour checker passport, using the adobe DNG editor to make a profile from the CCP file and then moving some of the points around until you get the colours that you want. Or you can use the LR HSL settings to change the colours to something closer to what you want, but that will provide a less accurate result.
If I've misunderstand any part of what you're doing, apologies