Convert hand caning to machine made

I would like to convert dining room armchairs from hand-caned to machine-caned. I need to cut a groove (about 1/8" wide) in the seat along the current caning holes, into which is placed the new machined caning and a spline to hold it tightly in place. I was planning to do this with with a laminate router. However, since the base of the router is too large to cut the groove in the right place (with the router held vertically), particularly where the arm meets the seat (only 1/8 - 1/16" from the edges of the current caning holes), I think that I would have to turn the router horizontally. I have not been able to find an appropriate bit or tool, or even to confirm that my thinking is correct regarding this process.

If the groove consisted of straight segments on the front/back/sides of the seat, instead of oval-circular, I might be able to use a standard slot cutter. But the groove really needs to be rounded to match the shape of the chair. I'm also concerned about the diameter of the bit, as I would need to turn the router as I go around the oval seat. I'd like to get a bit with bearings, if possible, to be able to control the position so it doesn't get away from me.

Any help would be appreciated.