Since your dream job appears to be with S&W, why don't you give them a call and see what they recommend, in terms of education and/or training? I've seen that happen many times where some aspiring person assumes a certain path to the job they want, only to find out that they've wasted time, money, and effort in the wrong direction. Smith & Wesson may selectively hire folks from one school, or not from another, or may not hire 'certified gunsmiths' at all.
I have spoken with a few smiths in recent years that have said the only education a gunsmith needs is in machining, then OJT working as an apprentice under a master smith. Most of these guys have commented that a school "won't hurt you, other than cost, but won't really get you any jobs except for at a place like Cabelas or Gander Mountain where you'd boresight rifles all day long".
Many states do not require any form of certification, license, or other education accolade for gunsmiths. From what I have seen - in researching this career path myself - is that many of the "schools" out there are very generically based, and tend to focus on bolt action riflesmithing, which would be wholly irrelevant for your target employer of Smith & Wesson.
The vast majority of folks that work at gun manufacturers like S&W are actually NOT gunsmiths. Specifically, the new product development teams are often mechanical engineers, and frankly, there's really not much getting developed on the older models, except how to produce them cheaper and more efficiently, which typically falls to industrial or manufacturing engineers. Smith & Wesson does have a "custom shop" where their in house gunsmiths amp up their production models, but I'd venture that those are very selective, very low turn over positions.
I'd tend to disagree that welding, machining, or woodworking courses would not help you, but I'd agree that you need to understand what kind of course material will be covered, compared to what you really want to learn in terms of gunsmithing skills. There are pretty generic woodworking skills that go into stock making: saw-mills, planes, jointers, routers, drill presses, and a lot of hand work. If you're using a machine duplicator, you're probably working at a stockmaker, not as a gunsmith, so that's kind of irrelevant. Welding would need to be TIG focused. Machining would be the most valuable experience, as learning to use the machine lathe and milling equipment is the more complex task out of the fleet. It's also the one universal component, whether you're blueprinting a remington 700, line boring a new cylinder for a S&W 629, or backboring a browning o/u. None of those courses would teach you how to apply woodworking, machining, or welding principles to firearms, but at least you'd know your way around the equipment, rather than showing up and trying to learn how to utilize grain orientation for product strength, how to use a router, AND how to route an action inlet at the same time.
Revolversmiths are a limited commodity, but a scarce few gunsmiths ever get rich even if they're one of a chosen few. Go in eyes wide open if that's the career path you follow.