"Does anyone have any proof that William Dungan, Rev. Thomas Dungan's father, is indeed a descendant of Charlemagne?"
The answer is no. No one has any documentation at all showing who the parents of William Dungan are.
There are plenty of hints as to what Rev. Thomas's maternal line (Latham) is, though there is a problem with his grandparents. Before that there is a very long purported line which, however accurate or inaccurate it may be, shows no sign at all of any royal ancestry. However, given the social stratum, the probability that Rev. Thomas Dungan is not in fact a descendant of some sort of king, be it Charlemagne, or a Scottish or Irish king, is exceedingly low. This is prime hunting ground.
Since there seem to be people on this list named Dungan who talk a lot about him, there should be male line descendants around. The first obvious thing to do in rummage around Ireland for male line descendants or cousins of the original (disproven) father of William and do a DNA comparison, and do the same for male line descendants of the other Dungan males running around the Strand when he was born.