The bay coloration would actually come from the Fjord as bay is a base color (brown dun). Fjords though are typically considered homozygous for Dun which dilutes the coat giving you the lighter presentation. Pangare being present lightens it further. A heterozygous Dun Fjord could exist and bred with a non dun breed like the Haflinger which are all sorrel then you have a 50/50 shot at Dun not being inherited leaving the base that was inherited (in this case bay) non diluted. As that is so uniform in their coloration because Dun is dominant you wouldn't visually be able to tell heterozygous from homozygous. If they did you could have the odd Fjord with the base color and no Dun but if they happen they aren't registered so though they would be Fjord they'd not have papers. I have friends that have said there were at some point quite a ways back non Duns but they are no longer accepted into the registry and haven't been for a while. Don't know how true that is but I have seen a picture of a horse claimed to be a non dun fjord on the net. It is an older photo. But that brings up the question of nd1. Could they also have nd1? With other genes present it may give the darker shades more of a Dun coloration as they are prone to fading and nd1/nd1 possible. Perhaps if there have been recent scientific papers published that would show whether it is in their genetic makeup.
Roaching is an option but not one that comes with a guarantee of timely regrowth. Some horses that are roached just don't get those long manes back. Slow growth and long turnover rates for that part of the body mean if you wanted the long look back you may be waiting for years. Those two things are genetic. As to thicker hair growing back in - the shorter the hair then the more follicles that have have hair of that length are exposed. The base of the mane is denser than the ends so you are just evening it all up.
Some horse people change their horse, they change their tack and discipline, they change their instructor; they never change themselves.