@ericmack94 You are correct that you cannot use the Acumatica app for this, and you are on the right track with the separate custom app.
On the products, unless the product is created as a Template item in Acumatica, the Variant logic will not kick in. The default variant that you see in Shopify is just a reference SKU.
The best option I see for your client would be the following: For the other products, you can ask them to setup the product with a different SKU but the same Product name in Shopify directly first. When the product sync runs it will look by the name but also respects the SKU.
The issue originates mainly when product is going out of Acumatica, but in this scenario since the same product name but different SKU is present already, it should create the reference links properly.
If you need more technical assistance specific to this customer, please feel free to DM me.