I concurrently found the exact same solution in Solidworks. Basically, all the algebra I had tried involved a circle tangent to the circular profile but centered around the roller axis (off kilter 45 degrees as well). The point on the desired curve is on this circle but straight above the axis. Since I couldn't do the math to find a continuous function to map it perfectly, I approximated with 4 points, actually drawing out the desired geometry and then creating a spline between the resulting points. It looked to be correct within Solidworks' polyhedral rendering error, so I decided to be satisfied. Here are some pictures:
Here is the initial cut of the elliptical roller (bumpy) with the circular profile.
This creates the profile surface to which the circles around the roller axis must be tangent. I then created four planes evenly placed along the roller.
On each of these planes, I sketched (shown next) a circle centered on the roller axis and tangent to the profile surface. The black line is along the profile and the circle is tangent to it. Then, on the circle I created a point straight up from the axis. The idea here is that this point, when revolved, will never go beyond the circular profile surface. If the roller profile is consists of such points, the roller will match the wheel profile. Here's the sketch:
I did that on each plane and got the four resultant points, all directly above the roller axis. I then created the following sketch, with a spline over the four points (actually I left the fourth out because I didn't want the roller that long), and then then rest of the roller shape. I mirrored it over to the other side to get a complete profile (and save work!)
And then I revolved it around the roller axis! (the centerline in the sketch)
The grey curve is the circular profile, showing that the roller does not extend beyond it.
Ta da! Hopefully this helps Solidworks people see what kamocat did. We had a little adventure working on this and hurting our brains together.
Edit: Sorry about the images taking so long. My internet is being quirky so I haven't been able to actually upload them yet.