“Rav Moshe says it is permitted to take public transportation if necessary”
Emphasis on the words, “if necessary”. According to Chazal if someone takes a route that involves seeing people untzniusly dressed and he had another option, he is a rasha.
In any case, the point of this discussion is to figure out the best option. If the whole purpose of his taking public transportation rather than driving the girl is to be more tznius and meanwhile it’s actually less tznius, that would defeat the point.