Advanced Shipping Manager App, Free shipping over $250, and product exceptions to free shipping

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We use the Advanced Shipping Manager App to get carrier rates for the products in our store. On orders below $250 - it will generate the correct carrier rates. For orders over $250 - the customer gets free shipping. However, our store has a number of very large and palletized items that are subject to extra handling fees and are not eligible for free shipping. My client initially set all these products up as exceptions to the free shipping rule - which does not allow customers to choose even a paid shipping option for these products.

I went into Shopify and tried to add our Advanced Shipping Manager rates to this "Custom Shipping Rate" - but nothing seems to be working. When these large items are added to cart, no shipping options are available. 

Is there a way that we can have these items excluded from the free shipping offer, but still have the Advanced Shipping Manager rates pulled only for these items? So shipping is free on eligible items and paid on ineligible?