My main reason for skepticism is that Roger C. Carmel was a year and a half younger than William Shatner, though he looked older. Harry would've had to be substantially older than his portrayer in order to have an adult daughter at a time when Kirk's nearly 26. I could buy that Harry was in his early 40s when we met him in 2266, about a decade older than his portrayer, but that would put him in his upper 30s at the time of Countdown to Darkness. It's not out of the question that someone that age could have an adult daughter, if he'd been under 20 when he conceived her, but that's kind of pushing the limits of probability.
This is a fair point. Carmel does look to be deep into his forties in TOS, but as you pointed out, the new film takes place even earlier than his original appearance. An adult-ish daughter does seem unlikely but maybe Bajor's sun hyper-ages children with human DNA (like it did to Molly O'Brien when she was brought to DS9) or perhaps Miss Mudd fell into that time vortex and came out 18 years old (again, like poor Molly did in Time's Orphan).