I've completed this project and wanted to document the technique so that others with a similar need can review the process that worked for me.
To begin with, I left the drawer front as an element and did not assemble it into a drawer until the carvings were firmly attached and cleaned up on the drawer front. The process resulted in full attachment and very minimum squeeze out. Clean up was trivial after the hide glue had fully cured. I did need to make some minor adjustments to the carving after the two symmetrical segments had been attached. It's surprising how minor differences came to exist even when I had made a good effort to shape them identically during the cutout and carving processes. I was glad to be able to wedge the front onto the bench while it was a flat board and not as part of a drawer assembly.
The process I used to attach the carvings was that described by former cartouche winner, Will Neptune. I did each piece of the carving as a individual glue up operation. First, the carving was located on the drawer front by a number (nine in my case) of wooden tabs that were 3/32-inch thick by 3/8-inch wide by 1.5 to 3-inches long. These were spot glued to the drawer front at the end of the tab furtherest from the carving. This makes removal and clean up of the drawer surface after the tabs are removed easy. The end of the tab that touched the carving was rounded so that there was minimum surface should glue squeeze out be an issue. I started with five tabs and continued to add them until there was absolutely no movement of the carving.
Then, with the carving on the drawer front, cover it with a single sheet of plastic refrigerator wrap. Then apply non-hardening artists's clay over the carving and when you think it is fully covered, place a sheet of parchment paper and a known flat board over the clay. Finally, apply pressure with a clamp. Remove the board and check to make sure the clay has been flattened by the board across the entire carving. Pay particular attention to all of the endings. Add small amounts of clay as necessary until you are satisfied that the board is applying pressure to the entire carving. When satisfied, place the clay on its pressure board into the refrigerator for at least 30-minutes to harden it. Take it out when you are ready to glue the carving onto the drawer front.
I deviated from Will's process in the method of applying glue to the carving. After experimenting with "stamping" the carving onto a flat surface containing a coating of glue, I was less than successful in getting consistent coverage. I opted to use a small natural bristle brush to "paint" a thin layer of pre-heated Old Brown Glue onto the back of the carving. This layer tends to sink into the walnut that I was using. I immediately "painted a second thin layer of glue onto the back of the carving and then positioned it onto the drawer front. I took a moment to be sure that everything was contacting the drawer front and then I applied the plastic wrap/clay/flat board over the carving and applied pressure with a clamp. I left the clamp in place for several hours and then removed it along with the board and clay. I let the carving cure overnight and then cleaned up the squeeze out with a chisel. There was little squeeze out, yet no voids under the carving. I consider this process successful.
I want to thank Will Neptune for developing this process and then explaining it to me. The process takes what would normally be a highly stressful, error prone operation, and makes it repeatable and reliable. Thanks Will.