I would use 3/16" to a max of 1/4" for that hawk and do a wrap and weld style. If done in 5160, there will be no need for a bit to be added. Forge the wrap around a 1" steel rod, weld the joint, upset and angle the face/cheek to start the heel/beard, draw out the beard, drift the eye, do final shaping on the grinder, etch. This will be a lot of work, but an experienced smith should have no problem.
If you have a power hammer and the tools ( and experience) you could draw out a piece of 1X3X3 into that axe.
All that said, the project you have drawn is a task for a very experienced metal worker and smith. The decorative work can be acid etched in for a rustic "Antique" look.