If you don't care about the noise...
I would get new bearings and races for the pinion, breaking a driveshaft hits them badly.
Install according to the manual, use marking compound for the pattern and gauge for the back lash.
What isn't in the manuals.. after you installed the pinion, use a dead blow hammer and tap the pinion good from the inside of the case and from the outside onto the shafts end. This is to reassure that all bearings and races are fully set. It does not damage the bearings. Check rotating torque after this again to see if it changed.
You'll always have noise from the damaged pinion and gears, even the smallest imperfections on the teeth generate noise. Yours are badly beaten up.
But it won't explode because of this as long as the bearings work.
But you would want to check the internals regularly for excessive wear, also if the noise changes instantly.
Have you ever considered getting another axle form a scrap yard and just replace the whole thing?