If he is the illegitimate son of Sarah, he would not be expected to match any Dyes outside his own small lineage. The test on ZKTR3 is just like that. This hints at confirming that Thomas could be the son of Sarah.
Barber is less clear to me. On y-search I was not able to identify any of the tested Barbers as being clearly descendants of Charles Barber.
If there is a direct descendant on y-search, then that hints that Charles is not the father - it is not rock solid proof because there could hypothetically be an illegitimacy in that line that would disguise the conclusion.
If there is no direct Barber descendant on y-search, we really have no basis for any conclusion.
Could you tell me the y-search code of the Barber test you are comparing so I could take a look?