Turns out the issue was not in the Ionic controller code nor the triggering html, but in the modal template, itself.
Per the Ionic documentation, I had each of my modal templates wrapped in this:
<script type="text/ng-template" id="repost-modal.html"></script>
By removing that and replacing it with just <ion-modal-view></ion-modal-view> the issue was resolved and the modals now open very nicely on the first click.