I wasn’t able to get anything working by only overriding the Ionic Sass variables. Although it seems like it should be possible as there is a $checkbox-md-icon-checkmark-style variable.
In the end I solved the issue by modifying the position of the checkmark using the following code in my app.core.scss file:
.checkbox-checked .checkbox-inner { position: absolute; top 9px; left: 13px; height: 11px; width: 5 px; }