When you buy the items in FRA ask for the tax refund paperwork before you pay. The staff is used to the procedure and will do it on the spot.

Fly to BRU, enjoy your trip. When you are checking in at BRU to start your trip to the US, you need to initiate the export certification.

If you are checking in the goods, the certification is done is BRU. So before checkin ask if the Belgian customs wants to inspect the goods before you check it in.

If you are carrying the goods in your hand luggage, you need to get the export certified at the last airport in Europe. There you need to present the goods to a customs officer after passing departure immigration.

The customs in any case will stamp the papers and you can cash the refund on the spot (the agent for the LH worldshop refunds is the 'Reisebank') or mail it in for a credit to your credit card account.