You are correct in theory. The document legally required is a declaration from your embassy that you have terminated your residency in the US. This allows you to directly import your goods into Spain without going through the UK.
In practice, we are US citizens and had no such document. When our shipment arrived (a full 20’ container), at first the customs in Cadiz asked us to get a custom’s import form from the head office in Malaga. But after much back and forth, they settled for a duty payment of ?570 and that saved us lots of hassle and paperwork. However, there is no guarantee they will always resolve the situation in this way.
Your only sure recourse to paying no import duty is to get that paper from the British embassy in the US. You had to originally register with the embassy when you arrived in order to get the termination of residence paper.