I've always repotted mugos in August, well after all new growth is hardened. Even at that, I lost one after doing some light root work and returning it to the same (relatively) shallow pot that it had been in for the previous three years. It suddenly started getting that lifeless grey overtone in the needles, not long after the repot. I lost another that I dug from the grow bed this past August. Otherwise, I've not been the least bit careful about the roots and the mugos have thrived (6 various cultivars of various sizes). The only thing that I can identify to have possibly been the cause is being potted deeply in relatively shallow pots.
My suggestion is that you keep the pot tilted15 to 30 degrees and return it to shade for a little while. Once buds stop withering, or it seems healthy despite the buds not moving, return it to full sun, but continue keeping it tilted. As long as the needles stay their normal bright green, it will be okay. Mugo needles last about 5 years and you can normally prune all the new shoot (just like JBP candle pruning) after it has hardened for 3-4 years in a row (it will back bud strongly). So, not to worry about the buds - worry more about the needle color. This may well not be what your tree is suffering from, but it will do no harm if I am wrong.
The catch-22 about root issues is that there is insufficient capacity to supply the foliage, so the foliage doesn't conduct as much photosynthesis, but this means lowered food/material and stimulus for root growth. So we must reduce the transpirational stress (shade and/or humidity tent) to get the foliage working, which then gets the roots growing, etc., etc.
Meanwhile, investigate the possibilities that pathogens involved further. Maybe someone else who has dealt with trouble similar to what your tree is experiencing will offer some insight into what is actually going on with your mugo.