Hey
Sorry jumping in here a bit late. Sounds like all you need is to bundle each of your product variants from the blank shirts. So you could have all White Size M shirt designs all linked to the same inventory SKU and thus once they sell whatever design it updates all your products.
We are testing a new bundling solution now if you are interested in seeing if it would work for your solution. There are other bundling apps available but most work around creating custom pages in a theme.
Regards
Tom