I'm going to assume that a diaper "cake" is a bunch of diapers [and related products] organized into a product that has the shape of a multi-level cake.
I think you've properly identified the key legal issue: Does the trademark first sale doctrine permit you to repackage genuine, branded diapers and other branded products into a composite product that you put together and then sell under your own brand. A good case to read about the trademark first sale doctrine is Sebastian Intern. v. Longs Drug Stores, 53 F.3d 1073 (9th Cir.1995) [ http://scholar.google.com/scholar_case?case=3440018830755182867 ].
No one can answer that question in the abstract -- your own attorney will have to look over your product, your branding, and your advertising. The general rule, however, is that, yes, it is lawful to resell a branded product SO LONG AS the quality of that product is not diminished and the mark used to brand that product is not used in any way to suggest an affiliation or endorsement or sponsorship of the re-seller [in this case, you]. As for the latter, selling your cakes under your own brand, adding a disclaimer asserting that your service is not related to the branded diaper, adding other differently-branded products into your "cake," even using differently-branded diapers in the same "cake" are all ways to dispel an association between your service and the company that sells the branded diapers. Again, speak with your own attorney.
FYI -- a case that unnecessarily threw a monkey wrench in the trademark first sale analysis is
Au-Tomotive Gold v. Volkswagen of America, 603 F. 3d 1133 (9th Cir.2010) [ http://scholar.google.com/scholar_case?case=6217578490156673480 ]. Your own attorney has to think through this wrinkle as well.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
The 1st sale doctrine is for copyright rights. It's also known as "exhaustion" because it means that once a copyrighted work is sold to someone, its copyright is exhausted and the buyer is free to re-sell the work to anyone.
Diapers aren't expressive works covered by copyrights, and neither are diaper cakes, whatever those are, unless maybe these "cakes" are sculptures made of diapers, in which case they could be the type of creative expressions that get copyrighted. But I doubt it.
This sounds like a trademark issue, and a question of how you can package products made by others into a new product constructed by you. I don't think you can, since these diaper manufacturers presumably don't make "cakes," and maybe they don't want to and don't want anyone to. They'd object to this business because their customers could easily mistakenly think that your "cakes" are made by them, instead of by you. This is expecially true if you only use one brand in each "cake," so that consumers will assume that the brand is the sources of the "cake."
But if you used several brands of diapers in each "cake," then it would more obvious that someone else, not the owner of any of the diaper brands, constructed the cake, and there'd be no confusion.
Avvo doesn't pay us for these responses, and I'm not your lawyer just because I answer this question or respond to any follow-up comments. If you want to hire me, please contact me. Otherwise, please don't expect a further response. We need an actual written agreement to form an attorney-client relationship. I'm only licensed in CA and you shouldn't rely on this answer, since each state has different laws, each situation is fact specific, and it's impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.
I wish to clarify one of the answers. The first sale doctrine is in fact a trademark law concept in addition to copyright law. Probably just semantics because as the learned response correctly states, the first sale doctrine is not an absolute defense against an infringement action when a branded product is incorporated into another product. In fact the same holds true in copyright law. Certain "combined" works can be deemed to be derivative works of the copyrighted work. Some courts have gone so far as to prohibit framing copyrighted art and selling it with the new frame (where the frame has some copyrightable aspect to it). Admittedly this is an overly legalistic answer to the question so I apologize. The short of it is that you have risk in doing what you want to do.