A friend wanted a whipped cream and cream cheese frosting for a birthday cake. I made royal icing flowers, since the frosting wouldn't hold up as flowers. The royal icing flowers dissolved into the frosting, even though the cake was refrigerated promptly and properly.
Any clues as to why this happened and any suggestions what I should have done instead?

The moisture in the icing is what killed your royal flowers. Royal and water equals trouble.
If you wanted to put the royal flowers on that cake you should've put them on right before serving or put some sort of protective layer between them and the icing. I've heard people putting melted choco on the backs of the flowers , raising the flowers off the icing with something like a little marshmallow or fondant, put the flowers on a plaque..or better yet use another medium for your flowers ie..gumpaste/fondant.
The same thing will happen again if you put royal flowers on a buttercream iced cake, so beware of that too. The royal will pick up the fat in the buttercream.
For this reason i seldom use royal icing flowers anymore. I like to get the cake decorated and be done with it.

HTH

You can use royal flowers on a crusted buttercream cake just fine - I do it all the time. (see my teapot cakes in my photos - all the flowers on the side of the teapots are royal) You just have to make sure your buttercream is crusted, and your royal flowers are very dry.

It does not work on whipped cream frostings, or on cakes that require refrigeration however!

My buttercream receipe does crust and the flowers months dry and still it picked up the fat..maybe it varies from recipe to recipe.