I would totally agree that the method Paul Miller outlines is one way to get the seamless approach and have solid wood for the pull, and this is the way I would approach it if the money was there, but not everyone has the knowledge, skill set, or tooling setup for veneering.
Back before the advent of commonly available vacuum press setups, when veneering was done either with a hot press or a veneer hammer, a lot of shops backed away from veneer work, or used contact cement for veneering, with disaterous results.
Many of the responses here are taking into account the original posters description of using pre-veneered sheet goods, and trying to put forth ideas that would achieve the desired result within those limitations.
In the past, I have rabbeted a lot of pre-veneered pieces, leaving 1/16" of the face, and gluing a block of solid wood in place to accommodate the shaping that required solid wood to be there, and edgeband the rest. It was somewhat labor intensive, but did not require a hot veneer press, so my employers at the time thought it was a good idea, as it also allowed grain matching the face panels without having a piece of solid wood breaking up the flow of the grain.
It can be done differently now that vacuum presses are available and fairly inexpensive, but the work would still be labor intensive.
That edge detail will cost a lot of money, no matter how it is approached. I hope that the original poster has enough in the budget to make money on the job.
Good luck.
TonyF