This is probably more than just a white balance issue, and related to the camera profiles your software is applying.
The SOOC jpegs are always going to be different, because each camera maker tries to out guess what kind of colours, contrast and saturation its customers will like and build those colours into the Jpeg style options. Even with raw files, every camera produces raws with slightly different starting points, and then you rely on how the software makers want to interpret those characteristics when applying their camera profile to the file.
However, the gey card approach might be a reasonable starting point, and you might not need to do this more than once. You can create a different preset for the corrections needed for each camera, and get LR to automatically apply the appropriate preset on import.. But a correction based on the grey card might not apply the correction needed for skin tones., or greens etc, because you are only correcting a single point. Nonetheless, it might be worth looking into before trying something more complicated.
You might not even need a grey card, just look at the raws as they open in LR and use the color calibration sliders to try and match the colour characteristics. You could also try the different profiles that LR offers. And then save whatever you end up with as a custom import for each camera.
You can also build custom profiles, which would make a better job of matching colours and contrast across the spectrum of colours (not just the midtones). The calibrite color checker passport will allow you to build custom camera profiles and there is also Lumariver
https://www.lumariver.com/ which is a slightly more sophisticated profile builder. You can also buy Profiles that are meant to be consistent across different cameras. e.g.
https://www.cobalt-image.com/.
DXO is meant to produce consistent results across all camera bodies if you use their custom profiles instead of the profiles they offer to emulate different camera manufacturer profils. I thought this was what the Adobe color profiles did, but obviously not and as I find the Adobe standard profiles rather unpleasant I haven't given them much attention.
I always start from a linear profile which is flat and fairly colourless, so the differences you are experiencing are far less noticeable, and I tend not to take photos that need colour matched. I'm not sure I've ever used SOOC Jpegs.
Maybe someone with more experience of dealing with the problem will be able to suggest a good solution.