After 24 hours I removed everything and it seems like the veneering process worked out. Some glue found its way through in a couple spots but overall it is pretty flat and well adhered.
I trimmed the excess from the edges with a utility knife and then sanded to blend it into the existing rounded edge. I'd say that I'm pretty happy with the result. Just have to give some thought on whether or not to try and even out the colour difference between the sides and the back now.