Discount across all variants of same product?

Indeed, the variants are handled separately on the checkout. It not possible to do differently.

Is there a reason why you're using variants for the product ? Because maybe you could do it differently:
You could maybe change the characteristic dropdown by a custom field of the table "item" (Business edition only) that you can create via the menu Display->Views. That way, you would be able to have a dropdown similar to the characteristic dropdown on the product page. However, the product information won't be refreshed dynamically like for characteristics/variants so, for example, you won't be able to display a different description based on the selection of the user. But that has the advantage that you have only one product, and the price quantities will be applied globally to the product regardless of the selection of the custom field.