Aw,boss,it's nuffink,wish that i could dedicate some time to this,as it's an interesting inquary,but the spring will now rush my ass out of the forge till probably fall-time...
Yes,the tooling is always a huge issue.It'll be a difference between night and day,to be properly tooled-up,vs dinking around with insufficiency of power And of tooling.
The way i tried it this brief time it was obvious that the metal will try to go out the sides(the path of least resistance).So before trying to indent the blade i first forged in a semi-circular indent in the edge of the blade.
Indenting the flat pushed it back out,as expected,and so i had to go back and forth like that,every other heat,edge/flat.
And yes,moving steel by hand is tough.I do come across a WI-body axe once in a blue moon,but you know how hard it is to molest a nice old tool,even it be shot for working with...I am rather rich in WI,and could,concievably,make a simple axe to experiment with...
Were you working without a drift in the eye? One of my concerns was the effort I would spend creating a drift before I could even attempt this. But if I controlled the heat near the eye perhaps the drift would be unnecessary.
It is a valid concern,as even in the best of circumstances the eye will probably have to be trued-up after everything's said and done.
I have not had a drift in(though i do have one for that size US axe),and Did have a stray blow or two fall close,so yes,it's a concern.
Drifts are damnable hard to make,they've some volume to them,and are quite irregular.Also it's of course next to impossible to have a "universal" one in any sense,so....
If you'll be making one,i'd try to make it as long as possible,both for the gentler taper,and in the handle.
A longish drift Really helps take that inevitable twist out in the end,an axe with no drift through it is Extremely hard to sight for twist...
And this brings us to tongs:Axes are notoriously hard to hold onto.And when holding onto the poll,it's rather easy to distort the eye.One would be better off forging a set to this purpose,i see people have success with a short/flat/stout-jawed kind.
And yes,seemed like all my efforts were beginning to add up to that very typical phantom bevel look/shape,(except that most of these you see are made in a closed die,and so don't have to conform to these laws that controlled hand-forging is ruled by).
Another necessary tool for this would be a Bolster block of N-th convexity,to forge the axe on edge.All i have for that is the base of the horn,and it's of course bad and wrong,horn being a dead-blow entity to begin with,And a degree of taper corkscrewing things on ya...
So,in an ideal world,had you some time at Larry's

,you could even make a combination tool-half bolster for edge/half mushroom for the indenting(to match some old 3-4 lbs sledge)...
It's not out of the question that in a moderately tooled-up rural forge they used to have something that could also have worked for this,some plow-share work for example also required convex bottom tools...