In boiler repair, they perform welding on the root of small schedule tubing by cutting "windows" out of the pipe. This would allow you to have access to the root but defeats the purpose of what you are trying to achieve as now you have to weld around the perimeter of the window.
Normally, small diameter pipe or tubing would be repaired by gouging or grinding out the defective weld area, and performing a new Complete penetration weld. Is this not possible for some reason?