You need to exert creative control over the product. If you created an appropriate image for the background, that would probably be enough. If you slapped the same decal on everything produced by an AI, that would probably not be enough.
Remember, AI generated work is in the public domain. So your question is identical to “Can I take a public domain work and alter it sufficiently to claim copyright on the product?”. The answer is yes, provided you make sufficient changes.