I doubt you have much to worry about. I cannot imagine the Mine Train having the same height requirement as Splash Mountain or Big Thunder, that would be absurd to me when both of those rides are much more thrilling and intense than Mine Train will be. From insiders here, Mine Train is going to be similar in thrills to the Barnstormer, just longer and with much better theming. The swinging cars are the only thing I could see having an effect on the height limit, though Lee here has said it's going to be a very mild and limited range of motion so shouldn't really be very intense at all. If this does affect the height limit, I would guess it's not going to be very dramatic. Maybe a couple of inches over Barnstormer. Though again I think it's safe to say it's going to be below Big Thunder and Splash.
So it's probably a safe bet to say it'll be 38" or below. There's no way I would imagine it requiring the same height limit as Splash or Big Thunder. We shall see though.