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I think they're in ultimate wilderness unless I've completely misread them. Specifically the parts on item and magic item salvage . . .
Raw Crafting Materials
Anyone trained in the Craft skill can salvage raw materials from equipment for use in crafting or repair. You must carefully dismantle the item to be salvaged, resulting in the item’s destruction. If the item’s price is 1 gp or less, its materials can be salvaged with only 1 hour of work; otherwise it takes 8 hours to salvage crafting materials. A successful Craft check against the item’s creation DC + 5 yields raw materials worth one-quarter the item’s price. If you fail the Craft check by 4 or less, the item is destroyed but the materials can still be salvaged in a future attempt. If you fail the Craft check by 5 or more, the item is destroyed and the materials are ruined. Salvaged raw materials can be used to create or repair any item of the same materials and reduces the construction time by the proportion of the new item’s raw materials that are salvaged (minimum 8 hours).
Raw Magic Item Materials
Anyone with an item creation feat can salvage the raw materials from magic items for the creation of new ones or repair of existing ones. You must have the item creation feat required for that item to salvage its raw materials. Each attempt requires destroying a magic item and 8 hours of work. If the item’s price is 500 gp or less, you can salvage its materials in only 2 hours. A successful Craft or Spellcraft check with a DC equal to 10 + the item’s caster level yields raw materials worth two-thirds the creation cost of the destroyed item (one-third the market price). If you fail the skill check by 4 or less, the item is destroyed but the materials can still be salvaged in a future attempt. If you fail the skill check by 5 or more, the item is destroyed and the materials are ruined. Salvaged raw materials can be used to create or repair any item made of similar materials or that shares any of the creation requirements as the original. Including the majority of the materials allows you to automatically meet any construction requirements of a new item that the salvaged item also required and reduces the construction or repair time by the proportion of the new item’s construction materials that are salvaged (with the usual minimum creation time). Spellbooks and formula books can be salvaged for magic inks and paper usable in formula books, scrolls, and spellbooks.
This is more for "raw materials" not "special materials" so its not directly applicable as your after the mithril not straps and the like. So I think your still in GM houserule territory even if its just a "yes i'll use these rules".
I think I'd require you to make the checks but if you succeeded you salvage the mithril to make the dragon barding plus a bit extra.
Mithril Half Plate I believe is 9,600 GP
A mithril chain shirt is 1,100 GP
Barding price is x4 = large, x2 = medium so a small creature is probably just base price of 1,100. Worst case is 2,200 for medium's x2.
By the ultimate wilderness rules a successful check would net you 2,400. So you can afford either one medium chain shirt barding + 200 GP Mithril to sell or a light chain shirt barding + 1,300 GP mithril to sell.
Which is why I'd be houseruling the salavging personally perhaps a sliding scale of success e.g. 2/3rd on a success, 1/4 on a failure so you're not worse off than buying/selling. Financially if you sell the half plate for half price as normal then buy the mithril barding you'll have more gold afterwards as your getting 1/2 value not 1/4. However you need access to forges and other equipment appropriate to the reclaiming so its more valuable the more valuable the item. Selling a 100,000 GP item gets you 50k, salvaging successfully gets you 75,000 and failing nets you 25,000. That way if you have the skills, time and resources it can be worth salvaging gear instead of selling it, if you don't have them then your better off taking the 50% purchase price. Also makes crafting a bit more worth it in the game as you'll slowly be getting better value for your loot.
As a pragmatic check compare any salvaging scheme to selling the object and buying what you want (or paying for the raw material for crafting), often seems to me that shopping is the superior value.
Unless Mithral is so rare it cannot be bought it would be better to simply sell the unneeded armor and use purchase the armor you want. A lot of the value of armor is because of the craftsmanship of the armor. The actual raw material of the item is probably only a fraction of the cost.
Raw materials are more than just the substance the item is made of. It also includes consumables that are required to create the item. Creating a metal requires more than just the raw ore. You need to be able to heat the ore to the correct temperature and often other materials are added to it besides the base material. Mithral and other special materials will have similar requirements. So even if you have an item of made of Mithral you will still need other raw materials to change it to something else. At minimum you will need a fuel capable of bringing Mithral to its melting point. Considering Mithral has 50% greater hardness than steel the fire to melt it is going to have to be a lot higher.
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While some of the crafting times are realistic for a Renassaince level technology, from a player's perspective they are generally unacceptable, unless you are an alchemist making alchemy items or some other class/archetype with a large bonus in crafting speed.
Even with a lot of downtimes, generally, there are better ways to make money.
Naturally, it is another matter if you want to craft for role-play reasons.
When it is part of the role-play, some of it can be done between game sessions, as Azothath suggested.