You’ll never drill a straight hole that long. Your drill bit will wander and come out the side.

Also, I think a steel rod, especially a round one, would have to be such a large diameter to prevent bending that it would weigh a ton.

I have rarely seen a vintage banjo neck that hasn't got some form of bow in it. Some have ben straight though. Remember, the forces that cause the bow started the moment it was strung up, even with gut strings.
I have worked on a number of very old banjos here in NZ and the majority I use my heating rig and clamps.
One particular banjo refused to unbend so I heated the neck up with my heating rig and proceeded to remove the fretboard starting from the heel. As I approached the 3rd fret, the fretboard suddenly sprung off and flew across the workshop floor. Upon inspection, I found that the 3rd fret was actually binding on the neck wood below the fretboard (it wasn't very thick) and was under tension until I released it. Glued it back on and bobs your uncle.

If it's a forward bow, and the fingerboard is hard and in good shape, you might could try compression fretting. But unfortunately some old banjos have fingerboards that are deteriorating to some extent. Probably heat treating is the best bet.

Taking off the fingerboard seems simpler to me than what you're suggesting. Then, if you want some reinforcement, a spline of carbon fiber could add stiffness without adding weight, and would require removing a minimum of the original wood. If you'd rout a channel big enough for a standard truss rod, IMO that would create weak spots at the peghead angle and the thumb string peg, such that a little ding in the wrong spot could cause a real beeotch of a fracture.

I have a fair bit of experience boring water mains in areas that can’t be open cut and then grouting them to keep them and the ground stable. It’s the same principle as you’re suggesting. The grout often doesn’t completely fill the void even after good planning and execution. I think you’re going to have the same issues. Plus we always expect the bore to veer off centre a bit.