Unless the designer owns the rights to the specific purse shape you use, or has copyrighted other specific elements intrinsic to the design, then it does sound like parody to me. It's a joke on "designer purses": it's not meant to try to trick people into thinking it is a version of a specific licensed product, nor are you using their copyright for your own personal gain. However the gray area would probably be: how closely does your version of the customer's initials resemble their logotype? Example: if I tweaked the Coke logo to say "Cole" and made a Coke-bottle shaped cake, they would probably easily win a dispute claiming I used their trademarked design (that would be: the logo, not the bottle). Because the Coke logo is a copyright, not just the letters c-o-k-e, but the design itself. MAYBE it would be ruled a parody, but I think that's a stretch. I'd actually be interested to learn more about cases like this where the "artistic rendering" was ruled to be a parody (or not).
Actually a better example would be: logos on purses. I don't think the Coach logo resembles the LV logo in the least. So putting a initials on a purse is not taboo. But putting initials that look like the trademark owner's initials...well that is where the problem lies.
Similarly (and further to your point) if you made Mickey Mouse with purple pants and green ears and called him "George the Mouse" it would be pretty clear that you are using The Sacred Mouse and trying to doctor him up. But I see your point: with a purse, if you are not specifically replicating a copyrighted handbag that LV produced, and you are not mimicking their logo, then it's simply: a designer-style purse with the initials "JS" or whatever. If enough elements are different, then it's not a knockoff, it's just a parody of designer purses. ANY designer purse.
BTW even though we've said you can't "just change" elements of a copyrighted image...in a way you can. I mean, let's say I made a mouse, and he just looked like a normal, photorealistic mouse. But I put red pants on him. Hm. I don't know if the Mickey's pants are a copyrighted element, but other than that, clearly my furry rodent with red pants can NOT possibly be confused with nor construed as The Sacred One. The closest it would be is...dum dum dum....a parody. You could even park him in front of a boat's steering wheel à la Steamboat Willy. Because the mouse does not resemble the cartoon, even if it does borrow an element of the cartoon (the pants) for humorous purpose. Add the white shoes and gloves and see how far we can take it before it becomes infringement .
Finally, back to one other point: I do agree that it doesn't matter that it's cake and not a real purse or product. Copying a TM design is copying a TM design, no matter the medium. Selling a cake with MM doing the grind with Minnie (oh, Debi!) is clearly a parody of the wholesomeness of MM, but it would still get you in a heap of trouble. But the heart of what you're saying is, you are not copying a TM design...am I hearing you correctly or should I just go crawl back under the rock from which I came?