IMPORTANT: THESE TESTS ARE ONLY VALID IF ALL THE TESTS ON THE SETTINGS PAGE RETURN SUCCESS.
1) Create a fake test product for like $1 with a SKU that is NOT in FBA inventory and set it to fulfill with amazon anyway. Purchase that and go through the full process. You should get an error returned from FBA to WooCommerce in the order notes (and by email) reporting a bad SKU. The test order will be set to the status "Fail to FBA" in WooCommerce. If that happens then you can be 100% sure your site is correctly talking to FBA and real orders with correct SKUs will definitely work.
2) For the full, accurate customer test, you can use a live product with inventory in FBA. We usually temporarily set the price to $1 or create a coupon for 99% or 100% off to use in testing. Purchase that product. It will go to FBA and create an order shipment. Then, you can quickly log into Seller Central, go to FBA orders, and cancel that order in Seller Central. Note that it might not show up right away in the list of orders in the queue. What we usually have to do is use the advanced search to return all dates / all orders from multi-channel fulfillment.
If you only do #1 you can still have a really high level of confidence that it will work. #2 is an exact real-life test. We've never had a shipment inadvertently go out before I canceled it. It can take hours even before it's past the initial stages.