The per-weight shipping module can be used for this.
1. Turn on the Per-Unit (perweightunit) shipping module in Admin->Modules->Shipping
2. Now manually edit *each* product and enter the shipping cost as the product "weight".
This module uses the product-weight as the shipping rate for this product ... ie: it is no longer the weight -- it is only the shipping cost.
3. Turn off weight display everywhere, since it is no longer meaningful.
a) Admin->Catalog->Product Types->[select your product type]->Edit Layout->Show weight->false
b) Admin->Configuration->Shipping/Packaging->Display Number of Boxes and Weight Status
c) Admin->Configuration->index listing / all listing / new listing / featured listing ... turn off the weight on these