Thanks for clearing your target, TJPaul.
This is not possible at the same time.
The asymmetrical work needs to be done after the symmetry modeling. Convert the symmetrical model with the Object Manager> Object> Current State To Object. Then Tweak any point individually.
Otherwise, the tweaks would be taking into account for the symmetrical work.
The following is more an illustration of the problems than a suggestion.
I don’t like even suggest this, as it is a tricky workflow and needs attention, while it might fail (!) along the way as well. So save often save early. I was able to make it fail from time to time when I lost attention.
Here is an experimental setup, and I sincerely mean it (I do not write that very often, so it is not just a hint), this is not tested at all, it follows only the request, and the Correction Deformer needs to be known very well, Read everything about it here:
https://help.maxon.net/us/#OCACORRECTIONDEFORMER
I think it is clear now that I take no heat for this if it goes wrong, tests this. No new points should be added after the Correction Deformer has been used, for example.
https://www.amazon.com/clouddrive/share/FEKpX9OtI171KNWNpzpQe8Yee2ov2OZW5lpzE1NssUn
Please note that any changes done on the non-symmetrical object might be canceled out by the opposite adjustment in the symmetrical tool. Which indicates one of the problems of your request. I stick with it, do the symmetrical work first, then use the unique adjustments.
You might suggest this need for a Sym-Asym Tool here:
https://www.maxon.net/en-us/support/suggestions/
For what it is worth: You might check how much the Sculpting tool. It might be the better way for your work. It has masking tools, to prevent accidents.
Enjoy.