I have been in contact over the years with other (proven, they say) descendants of Sir John Yeamans...and it sounds as though you have a pretty good case...why don't you put out an SOS...and get your lineage to him verified...others have and I'll bet you can, too.
If you want to e-mail me your data (privately), I will be happy to put forth some research effort on your behalf...even if your ancestor did poison my ancestor!!! (I don't believe in continuing family grudges - and you seem like a nice sort...)
With this "ancestor hunt" we must ALL be able to take the good with the bad...
By the way, there are at least two books relating to this situation that go into detail - so I don't think it is fair to blame Col. Cave when the questionable death of Col. Benjamin Berringer is a matter of history and public record.
Would love to see that miniature/and think it is exciting that you have it.
At one time - and they may have this - as a permanent display in Charleston, SC - there was a display of lovely early miniatures which I found most interesting...some were of the early settlers of the city, etc. Perhaps you could contact the Gibbes Art Gallery about your miniature. They would be delighted, I am sure, to hear of it. Or, even better, send them a close-up picture.