Hi Jeremy -
I may not understand exactly what you are trying to do - but I do have a "sandbox" cart where I experiment. I dropped your code in several places in product.tpl and it worked as far as displaying the desired extra field. I don't have a tab mod, so that may be why I can't see the problem you are having.
For your other question - was this what you need?:
Code:
{section name=field loop=$extra_fields}
{if $extra_fields[field].field eq "Ingredients"}
{if $extra_fields[field].field_value ne ""}
{assign var="ingredient" value=$extra_fields[field].field_value}
{/if}
{/if}
{/section}
Then - you can use later:
Code:
<p><b>Ingredient:</b> {$ingredient}</p>
If it helps - you can PM me and I will give you access to my sandbox.