How to avoid "Double-instantiating a pane on an element"
I was able to resolve this by testing to see if the pane existed before re-instantiating the element. Happy to consider another answer if someone can explain what is going on better. For now, the following worked:

// Check to see if an element was already registered as a `Craft.Pane`:
if (!$("#myelement.pane").data('pane'))
{
    // Create a pane object if it is not
    this.$pane = new Craft.Pane($("#myelement.pane"));
}
else
{
    // Use the existing pane object if it exists
    this.$pane = $("#myelement.pane").data('pane');
}