Why would Muslims engage in commerce with non-believers of the West?

Because muslims, just like non-believers, also benefit from trade.

Explanation:

The concept of free trade is built upon the concept of comparative advantage. This means that muslim countries have specific economic areas where they are more competitive, and by focusing on these areas, exporting the surplus, and importing what they do not produce, they gain a net economic benefit.

This alone is a motivation for muslims to trade, whether with other muslims or not.