We do have republican chaps in Parliament. But the OP isn't suggesting that people are suspicious of Harry's parentage as IOTL, but that it is common knowledge that ATL Harry is not Charles' son. If such a situation comes about, I think it's pretty plausible that ATL Harry's going to end up out of the running.
I'd agree; most likely Harry would voluntarily concede that he's out of the running once his parentage became public knowledge in exchange for still enjoying a small amount of royal patronage.