The most elegant and fool-proof solution would be to build a custom fixture to hold the neck and guide the router. How you do that is totally up to you....and that's the hard part. You need to invent it. The major hurdle may be just getting your head around the idea that building a suitable fixture may take quite a bit of time, you may need to buy materials, and you'll need to spend money. You might spend a day building the fixture, and then spend 30 seconds making the cut. But you will know the cut will be correct, the first time.
But you may not have to do that. When I replaced the truss rod on my Harmony 1260, I was able to pull out the old truss rod without removing the fretboard. I was then able, with a lot of fiddling around, to pull a homemade rectangular broach through the hole and slide in a new truss rod.
The thing to take from the way I did it, I think, is that you don't need to take off very much. My homemade broach idea was capable of only minimal wood removal. The existing channel is pretty close to the size you need already. Instead of using a router, you might be able to do the job with a narrow chisel, cleaning up the existing channel. I don't know what the dimensions are of the truss rod you bought. I used one of these:
https://alliedlutherie.com/collections/ ... truss-rods
which are 1/4" wide by 3/8" high. (They are also beautifully made).
I think before I went through all the effort to build a routing fixture, I'd try a 1/4" chisel.