Dwarves learn things by doing. The fear of migrants is something I'll never understand. More dwarfpower is a good thing. Likewise, I'll never understand the mindset that a dwarf that arrives with labor X must do labor X forever. Dwarves are adaptable.
Do you lack masons? Turn the fish cleaning labor off on a dwarf, put him as a mason, and set his custom job title (with y) as "Mason". He will do mason-y things and learn how to be a mason. He'll get pretty good at it, too, if he's the only mason who's building things.
Right now, dwarves are able to learn skills at an insanely fast rate compared to real life. Without a teacher, in the real world, it takes a lot more time to re-invent a skill yourself, but a dwarf is apparently supplied with the inbuilt knowledge of how to engrave pictures, make complex mechanisms, construct furniture or hollow mugs or blow glass or farm plump helmets from birth. Use it. Put dwarves to work outside their specialties and don't worry about it in the slightest.
Any migrants you don't have a use for, just put them to use in an area which is undermanned. Likewise, if you have a shortage in the militia, simply change some poorly skilled dwarf's profession. It is entirely valid and not "wrong" for you to conscript random farmers into the militia in a time of crisis or to replace losses. You have to use what you have.
Likewise, for a militia, you don't even need weapons. Dwarves can kill snatchers by dogpiling on them and wrestling, and sometimes wrestling can be quite surprising. If you are backed into a corner, don't just watch as death comes. Have your dwarves fight with their bare hands. While you build them simple copper spears (spears are some of the best choices for weaponry with poor metals, for a variety of reasons) they can spar unarmed and build up the wrestling and dodging skills they need later. Dwarfpower for your militia is not a bad thing, because, again, dwarves learn by doing. Change their custom title to "Militiaman". Put them to work learning how to wrestle. Then put them to work learning how to stab things.
Think like a military commander, not like a matchmaker in a feudal Asian society. Both manpower and skilled labor is important. Work to develop skilled labor, but do not be afraid of cross-training. Specialization is for insects. Your dwarves can serve any labor you put them to.