Say this girl is three weeks into her pregnancy and an emotional basket case. Two things of note regarding your hypothetical: 1) at three weeks, the 'girl' wouldn't have a clue that she was pregnant; 2) by the three weeks you mention, for the new individual human life in her womb, the alive individual at his/her normal life cycle stage has a beating heart that is developing chambers already, so anything to discourage slaughtering that new individual life outweighs the 'possible' publicity manipulation the pro-serial killing crowd might raise. Protecting the innocent and giving the girl a chance to choose life instead of killing are the central issues.
Full disclosure: I live within 19 miles of Bristol. If the parents approved the drive to the Virginia side (or from VA side to TN side) and the brother drives her there, it is highly unlikely that the issue you have created would even exist, especially for a three week gestational age pregnancy. Such creation of specious hypotheticals does nothing to forward a reasoned debate on the real issues that will arise over such bills as this wise young man has written about.
We're on the same side here. I want to reduce the numbers of abortion in this country too. It's just that I think we have to pick and choose our battles.
Would banning people from crossing state lines stop a few abortions? Yes. Would the inevitable "hard cases" create negative publicity for the pro-life side and thus prevent victories elsewhere? Yes.
Realistically, the only way to significantly reduce the number of abortions in this country would be to convince people that having abortions is a bad thing. If you pass a law that half the country opposes and a minority strongly opposes, all you have is an ineffective law.