Yeah, so if this is a standard 12v wire you're running, try prying off the door trims. I'd bet good money there's at least a half inch of space somewhere in there you could run this wire through to the outside, possibly leaving a small connector inside the wall that you just plug the solar panel and battery into when in use. Something like https://www.harborfreight.com/solar-power-connection-cable-kit-59171.html one of these kinds of connectors. Then just carve out a tiny bit of drywall to allow the trim to sit flush, recaulk and you're in business.

Can't tell you that without a diagram or putting eyeballs on it myself. I was having a hard time following your description as it was.

I suspect that the red wire goes to the hot input for the pathway lights that now don't work at all. First, check the red wire at the sensor and confirm it does not have AC voltage present when disconnected, then pull the nearest light to the sensor and I bet you'll find the other end of that red wire. A quick continuity test will confirm that.

That being the case, just wire that red line in with the sensor's output and they'll light up just like they used to.

Now, that all changes if you have AC present on that red line with it disconnected, because then they ran 220v to it for who knows why. I strongly suspect they just ran the 220 wire because they intended to use the 4th leg as a separate switched circuit kind of thing, but still on 110v

https://www.harborfreight.com/30-watt-5-in-1-hobby-woodburner-63850.html

Hey, look at that, $10 for a heated x-acto holder.