How to machine this part?

@MrWhoopee's solution, a woodcutting router bit should work, if you rough it first with a ball endmill so you're not taking more than a .005 or so cut. I'd certainly try that first.

Another option is to rough it out with a ball endmill. Then turn it up on its side so the long axis is aligned to the X-axis on your mill, and the open face is facing to side (not up like shown). Then use a fly cutter with a ground HSS form tool matching a section of the 1.125" curvature. You'd need to hand grind a few HSS form tools (depending on the width of the tool blank). You can flip your form tool over, run your mill in reverse, and do the other part of the radius (top vs bottom).
This definitely sounds like the best path for a small mill. Great recommendation rabler Clear the center with an end mill then Calculate the coordinates for a series of small steps staying say 0.020” away from the profile. The fly cutting tool will need to be ground with a small radius at the edge of the main shape so you can blend it. Again calculate the coordinates for each step. Do one half, flip the part, indicate it back in and do the other. its possible. Will take some time for sure.
 
You'd be amazed at what a sharp D-bit can do. Grinding the outside radius is low hanging fruit. You should probably rough cut first to remove material. HSS or even O1/W1 work fine in 1018, if that's what you're going with.
 
How big is your lathe?

Attach it to something your lathe can hold.

Make a large round object.

Pie cut after done to make both halves of the die.

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You can also try something like this. I didn’t see an 1-1/8, but with 1” it should work with blending the steps.


I’ve used some of my woodworking cutters on steel and they work well if you take small cuts.
 
You'd be amazed at what a sharp D-bit can do. Grinding the outside radius is low hanging fruit. You should probably rough cut first to remove material. HSS or even O1/W1 work fine in 1018, if that's what you're going with.
I was in the middle of composing a reply about this when the router bit occurred to me.
 
View attachment 421834

Almost sure to be carbide . And so cheap you should at least try it. I would try to cut it from scratch. It should work if your mill is stout enough to handle it.
 
In my experience, the weak point to the woodworking carbide cutters is the shank. They tend to be some crap-cast metal about 1/2 the time that just fractures if you run it too fast. So make sure you keep your feed rate low enough/dont 'push' the cutter in too much.
 
Carbide tipped router bits will cut 304 SS just fine, so should have no problem with 1018. That's a big bit and is going to take a rigid setup to run it. As others have said, I would hog out as much material as possible with a much smaller flat or ball nose end mill then carefully run the finishing passes with the router bit.
 
For short run projects I used high density mould foam for the former and Delrin for the long former. It formed 28mm Copper tube at 250mm radius for 180 degrees. It worked great for my coffee table base.28dia 250 rad1.jpgCoffeeTable_1(BLACK)2.png
 
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