Just thought I would put this here for anyone else who needs a solution like this:
Inside your function (that hooks to the asp_stripe_payment_completed hook) you can use the conditional tag:
if ($post_data['item_name'] == 'NAME HERE'){ //do this }
where the NAME HERE matches the name='NAME HERE' parameter in the [accept_stript_payment] shortcode.
This can be useful for all those “non-product” based stripe buttons you may need to create dynamically.