I see apples and oranges in this question. The chassis is cool, but you it is ONLY the chassis and does not include an action or barrel, so it doesn't really play any factor as to whether you can swap the barrel or not. The only factor is if that the chassis can handle the outside diameter of the barrel, which has more to do with the design of the barrel rather than the caliber.
The Savage 10 Short Action is available in multiple rifles styles in multiple calibers, including .308, .243, and 6.5 Creedmoor (all use the same bolt and magazine). It would be expensive to purchase a .308 Win chambered rifle and then swap out the barrel for a 6.5 Creedmoor chamber--and swapping the barrel on a Savage 10 bolt action rifle is not quite as trivial as doing so with an AR-style--you will probably need a gunsmith to do it. Just get a Savage 10 SA-based 6.5 Creedmoor rifle on sale somewhere and swap the stock for your chassis.
Also note that the Model 10 SA actions have come with two different spacings between mounting bolts on the bottom (4.275" and 4.4"). You will want to make sure the chassis fits the rifle.
A .30 caliber suppressor will work with a 6.5mm barrel/cartridge with likely little discernible difference than getting a 6.5mm suppressor. The Saker is rated for .300 magnum cartridges, so I would not worry about shooting 6.5 creedmoor from it.