Nice design BTW.
import the poster design itself into Blender, and export it back out again?
Yes. So there is something to refract.
When importing your bitmap as planes, remove the BSDF node, and connect the image node to the output node directly. That way the colors will be preserved, protected from light falloff, and without the need of making the material as an emitter.

I just feel that importing the poster design into Blender, and then exporting it back out... will have some kind of negative impact on the quality of it?
Basically yes. The vector elements are not vector anymore.
But you must have in mind an important concept in 3D rendering. Passes.
So, make an additional pass assigning an opaque material on the ice cubes and render a mask. You can delete all other elements and render a transparent PNG and extract the mask from there, or make a black environment and add white emitter material on the cubes. Now you have a mask. (You probably need to adjust the levels to have pure white and pure black)

You can use that mask to cut the rendered cubes and put the new raster image on top of the original design.
You could move some vector elements, like the text over the raster images. (But with your specific design, you will probably have issues using vector elements over the ice. The mixing could look artificial. In some other designs, try to keep the text as a vector).
If you need shadows interacting with the background image, I would also make them on a different pass. Mask and merge them using the multiply blending mode. Play with the levels to make shadows stronger or weaker.

With your 3D elements now as layers, you could rasterize the original background poster at 300PPI and then the rasterized cubes as a new layer. That way the original background design is only "rasterized" once, not having the additional blurriness of the render.
One additional important thing to do is change the Filmic color profile to Standard. That way the RGB original tones of the poster will be preserved.

P.S.
But also, a faster method would be that you export your original design at 2x, and apply some sharpening to the Blender render. (Just change the color profile as mentioned) But only if the original design is also in RGB mode. If the original design is on CMYK mode you need to composite.