[...]what I meant Eowyn would have won the skirmish and Shagrat lost it, but does he not get to place his exertion?

Eowyn wins the skirmish because no minion is left after the skirmish had started. If Shagrat is killed, he can not use his ability anymore, because he is already dead, before he looses his skirmish. Eowyn is still ther, so she is physically be able to be the winner.