Any chance at getting some laser output specifications other than 20W? Is it pulsed or cw? This will assist in understanding what the fiber is. Fiber laser resonators are almost always start single mode, ~10um core. A larger core output fiber then can be bonded on if that’s what the intended use needs. It’s also important to know what the fiber is that failed. Is it clearly the output fiber or does it connect between internal pieces?
Buffer is a hard polymer coating over the clad layer, (like shellac on a transformer winding). The 125um figure sounds to me like that includes the buffer. The 250um must be some jacket material. The jacket may be any old tubing if it’s just protecting the fiber inside the laser package. One of the areas that require practice is the safe stripping of the buffer layer. Kits generally provide a tool but a pocket knife can work just fine. The telecom fiber you mention will get you started. The bond quality is then qualified by some bend and tension tests. We can talk about this when you get a little further along.